On the evening of Thursday, June 26, the Pentagon sent a bold hostage rescue plan to the White House for approval. Dozens of Special Operations forces would fly into Syria under the barest sliver of moonlight, set down in the heart of Islamic State territory and snatch four Americans being held by the militants.

The landing took place almost exactly a week later. Yet the commandos who rushed through gunfire into the makeshift prison found only half-eaten meals and a wisp of hair. The hostages had been there. But they were gone.

By the end of the year, the Islamic State had brutally killed three of the Americans, posting their videotaped beheadings online. The last hostage and the only woman, Kayla Mueller, was declared dead last Tuesday after the militants sent photographs of her body to her family.

The finality of that news has given rise to painful questions about whether more could have been done to save them. Grieving relatives of the victims, some of whom have accused the administration of waiting too long to launch a rescue mission, have also criticized the U.S. policy of non-negotiation with hostage-takers.

The administration has acknowledged that it could do better at communicating with the families, notwithstanding its ongoing rejection of paying ransom, and has launched a review led by the National Counterterrorism Center. “We will do everything we can, short of providing an incentive for future Americans to be caught,” President Obama said in an interview last week with BuzzFeed.

Scores of hostages, including Westerners, have been killed by the Islamic State since 2014.

But some of those who worked on the rescue mission say they think the White House itself is at least partly to blame for the failure. They charge that there were delays in bringing the plan to Obama’s desk and that, as a result, the rescuers missed the hostages by a matter of days, or even hours.

In interviews with The Washington Post and in other published accounts, a number of ­operational-level U.S. intelligence and military officials, speaking on the condition of anonymity to voice criticism of ­higher-ups, have said their disappointment at the failure of the mission was mixed with frustration over the decision process.

Obama himself acknowledged in the BuzzFeed interview that they “probably missed [the hostages] by a day or two.” But, he said, it was inaccurate “to say that the United States government hasn’t done everything we could.”

Senior administration officials strongly denied there was any delay in the approval, especially once it reached the White House.

Instead, four senior officials directly involved in the decision, and several others with close knowledge of it, said that one of the most complex and dangerous such efforts ever undertaken had moved through the planning, approval and execution process at what one called “warp speed.”

To prove their point, the officials revealed new details about a rescue mission that, once it failed, was never intended to be made public.

“For us, the clock starts when they tell us they have an operation that they want the president to review and approve,” said ­Susan E. Rice, Obama’s national security adviser. The clock on the mission to rescue James Foley, Steven Sotloff, Peter Kassig and Kayla Mueller “began on a Friday and ended on a Saturday evening” when the president, meeting with his top advisers, gave the “go” order.

“It can’t happen any faster than that . . . particularly given the complexity of the risk,” Rice said.

A senior Defense Department official said he understood the frustration of intelligence operatives, planners and “the guys ready to go.”

But this “was a risky operation, deep into Syria, where we hadn’t been before,” he said. “It involved a lot of people,” with substantial danger to both the troops and the hostages themselves. “It wasn’t a nice little surgical operation.”

Moreover, while the rescuers found evidence at the site that the hostages had been there, the mission was launched with no definitive intelligence confirming their presence.

“There wasn’t a hot, smoking trail,” the Pentagon official said.

Deputy White House national security adviser Ben Rhodes acknowledged the feeling that “if the timing had been different, maybe the outcome would have been different. The fact of the matter is, if you’re talking about a major military operation inside Syria, the notion that you shouldn’t have any review of the plan before it goes to the president for signature is contrary to how things should be done.

“Frankly, the president made an in­cred­ibly risky decision,” Rhodes said. “In some respects, it was more risky than the bin Laden operation.”

Hostages in Somalia, Yemen

“Hostage rescues are the hardest thing we do,” the senior Pentagon official said. “It’s twice as hard as capturing a prisoner” in a hostile environment overseas, as Special Operations forces have done successfully on a number of occasions during the Obama administration.

If the target prisoner gets killed in the process, he said, “it’s no big deal.”

Osama bin Laden, whether or not it was part of the plan, ended up dead. Terror suspects captured by U.S. raids into Libya — Nazih Abdul-Hamed al-Ruqai in 2013 and Ahmed Abu Khattala in 2014 — both ended up in U.S. prisons.

But a rescue mission is a different story. According to a range of officials involved in such actions, the factors weighed begin with the level of certainty about where the hostage is located. There was only one case among attempted and successful rescues in recent years, officials said, when there was absolute knowledge of where a hostage was and the conditions of captivity.

The January 2012 rescue of American aid worker Jessica Buchanan and her Danish colleague was considered a relative piece of cake. With substantial overhead and ground intelligence, Pentagon officials assured Obama that they were completely confident of her position in the wilds of north-central Somalia. They knew that the al-Shabab fighters holding her were lightly armed and poorly trained and that there were no reinforcements nearby.

A team of Navy SEALs dropped into the camp. One jumped atop Buchanan to shield her, the others killed a number of her guards. The successful operation was over in minutes.

Last fall, long after the Buchanan operation and the failed summer raid in Syria, U.S. commandos made two attempts to rescue American photojournalist Luke Somers, kidnapped by al-Qaeda in Yemen in 2013.

In planning for the first attempt, in late November, officials thought that several hostages were being held in a cave, although they were not sure Somers was among them. There was no information that his life was in immediate danger, but the location was considered easy and the risk to U.S. forces was judged to be fairly low. The operational plan was presented to the White House on a Thursday; Obama approved it on Sunday.

All of the presumed terrorists at the site were killed. One of the rescuers was wounded in the arm. Somers was not there.

Less than two weeks later, new intelligence located Somers in a compound of buildings in the southern Yemeni province of Shabwah, and there was a “clear indication he was about to be killed,” the senior Pentagon official said. Intelligence on the location was good, although the assessed risk was high. Significant components of the previous attempt were still in place, and Obama’s approval came within a day.

The Dec. 6 mission resulted in the deaths of Somers and another hostage the Americans said they did not know was being held with him, South African aid worker Pierre Korkie. Both were shot by their captors, who were alerted to the landing force before U.S. commandos could reach the site on foot. Korkie’s family and employer later said the failed rescue mission came just as they were negotiating his release.

Intelligence on Americans

In March and April of last year, Islamic State militants in Syria released at least six European hostages. Although the exact circumstances of their freedom remain murky, most if not all are presumed to have been exchanged for ransom.

At least some of them had been held with the Americans. The Islamic State frequently moves its captives, but after debriefing the released Europeans, U.S. intelligence officials believed that at least Foley, and possibly more, were at a location outside the militants’ de facto headquarters in the north-central Syrian city of Raqqa. On that basis, the Joint Special Operations Command, or JSOC, began drawing up a rescue plan.

The hostage site was described as a compound of buildings, including a small oil refinery, on the city’s outskirts. A substantial concentration of militant fighters was said to be nearby, along with the main Islamic State force in Raqqa, whose prewar population totaled about 220,000 civilians.

At the time — months before U.S. airstrikes would begin in September — there was virtually no air surveillance over Syria and little desire to risk engaging Syrian government air defenses. Information from the released hostages, the administration believed, was inconclusive and intelligence from inside the country, particularly from areas wholly occupied by the Islamic State, was scant at best. The Tampa-based Central Command, presented with a preliminary proposal by JSOC, was hesitant.

Meanwhile, the hostages’ families had been gathering their own intelligence. To them, the situation seemed far more clear-cut than it did to the administration, which they believed was dragging its feet.

Some remain bitter. “Nothing was done to save our young Americans,” Diane Foley said in an ABC News interview last week. Her son and the others were “held for nearly two years, and there were many opportunities along the way: several times when the captors reached out, several times when returning hostages brought sensitive information,” she said.

“It is no secret that the administration neglected this case,” Barak Barfi, the principal hostage adviser to the Sotloff family said in an interview. “It never dedicated the necessary resources to locate the Americans. . . . The administration was always two steps behind us as we were the first to debrief European hostages as well as receive information from Syrians who had been held with the Americans.”

For the administration, the breakthrough that pushed the rescue plan forward came in mid-June with the release of another European, Danish photojournalist Daniel Rye Ottosen. After a ransom of what Danish newspapers said was about $2.3 million was paid, Ottosen was taken by the Islamic State from the Raqqa prison on June 17.

He was released from a militant safe house near the Turkish border on June 19, and on June 22, FBI agents interviewed him in Denmark. Ottosen helped solidify the information learned from the other released hostages. He had been with all four of the Americans just days before and provided a description of the prison site so exact that planners had enough certainty of where it was, and what the rescuers would encounter when they touched down, that the plan began to move up the military chain of command.

The Central Command signed off, despite ongoing doubts about the quality of the intelligence, the large number of troops and other assets involved, and the fact that Syria was unknown territory, administration officials said. From the Central Command, it went to the Joint Staff, and then to Defense Secretary Chuck Hagel, who signed it and sent it to the White House.

The raid into Syria

The plan, by all accounts, was unprecedented in its degree of difficulty and hazard. More than 100 Special Operations troops were to be involved, along with an array of fixed-wing aircraft and helicopters.

“It required more resources, more effort, more time to stage and prepare” than any previous such operation and involved representatives of all U.S. military services, a Pentagon official said. “It was so freaking complicated.”

Because of the presence of a large force of Islamic State fighters, this official said, the plan involved several “diversionary elements” designed to “distract the handlers from the site.”

“We had to have the assistance of some countries in order to stage assets,” a senior administration official involved in the process said. “That is not a decision that the Defense Department or JSOC can make.” Other officials said the operation was launched from Jordan.

“We had information that made us think they could be there,” the senior administration official said of the Raqqa prison site. “But we also didn’t know what they were walking into. We didn’t have a huge intelligence base. We didn’t have overwatch. We didn’t have a lot of what you might have in other operations.” The risk to both the hostages and the commandos was assessed as high.

For the Special Operations forces readying to take action, the wait was nerve-racking. “These guys are snake eaters,” the Pentagon official said. “They move quickly; they don’t want to be slowed down for anything. They don’t live in the world” of policy decisions.

Senior representatives from all national security departments and agencies convened at the White House early on Friday, June 27. Later that day, their Cabinet-level principals met to examine the plan so that it could be presented to the president with all questions answered and risks assessed.

On Saturday, it was forwarded to Obama, with a recommendation for approval, by his top national security team. “I frankly expected the president to say, being presented with such a high-risk, complex operation, okay, thank you, let me sleep on it,” said a senior official who was present at the meeting in the White House situation room. But Obama signed off before the meeting ended.

Presidential approval set a series of events in motion. Jordan needed to be briefed on the final plan and agree to it. Special Operations forces assigned to the mission had to be gathered at their home base at Fort Bragg, N.C., and transported to the Middle East. Aircraft had to be positioned to carry the rescuers in and out, participate in the diversions and conduct overhead surveillance while the operation was underway. The weather and phases of the moon had to be studied, to ensure cover of darkness.

The commandos would fly in heavily armed Black Hawk helicopters from the Army’s 160th Special Operations Aviation Regiment, known as the “Night Stalkers.”

When the force landed in the early morning of July 3 in Syria, their stay was brief. What one military official said was “a good number” of militants were killed. One U.S. service member received a minor injury when Islamic State fire hit a helicopter.

There were no hostages. Despite later conjecture that they might have been moved only a few hours earlier, and Obama’s estimate of a day or two, several senior administration and military officials said it could have been days, or a week or more. “We don’t know,” one said. “We know that they were there, based on what we found. But we don’t know” when they left.

News of the raid, and its failure, would not become public until Aug. 20, the day after the Islamic State posted a video of Foley’s beheading.

A similar video of Sotloff’s killing was posted Sept. 2, followed by Kassig on Nov. 16. Last week, when Mueller’s death was announced, the administration confirmed that she, too, had been one of the four held at the prison outside Raqqa. U.S. and Jordanian officials denied that she had been killed, as the militants claimed, by a coalition airstrike.

“I understand the frustrations of folks who are working their hearts out, removed from what goes on in the Pentagon, in the White House, in the State Department. I get that,” one senior official said of the failed rescue mission.

“I don’t think it was too long a time,” the official said. “I think it is heart-breaking that they weren’t there.”

An earlier version of this article misstated the date the U.S. force landed in Syria. It was the early morning of July 3, not July 4.

Adam Goldman contributed to this report.