President Obama stands with Health and Human Services Secretary Kathleen Sebelius in the Rose Garden at the White House on Oct. 1. (Charles Dharapak/AP)

As the kinks in the government’s health-care Web site have multiplied, the question being asked across the partisan divide in Washington is: Who will the president fire?

The answer for President Obama is that punishing underlings for administration mistakes is not as simple as it may seem.

Cathartic, perhaps. But dismissing Cabinet secretaries, senior agency officials or those even lower down the bureaucratic ladder carries political risk for a president — especially one who hopes that the technical flaws undercutting his signature domestic legislation will be fixed before the public relations problems get much worse.

“Generally, presidents are very bad at firing people,” said Stephen Hess, a senior fellow at the Brookings Institution who advised four presidents. “These things happen. But it reflects a mistake the president has made, and they don’t like to admit mistakes.”

Increasingly, Obama is being asked to do just that by lawmakers on both sides of the aisle. Last week, Republicans began calling for Health and Human Services Secretary Kathleen Sebelius to resign or be fired, and now Democrats are joining the someone’s-got-to-go chorus.

Sen. Bill Nelson (D-Fla.) issued a statement this week calling the troubled rollout of the Affordable Care Act’s federal Web site “in­excusable” and added: “Somebody ought to get fired.” On Wednesday, Rep. Rick Nolan (D-Minn.) was even more direct, telling reporters that Obama should “man up, find out who was responsible and fire them.”

Obama is by political nature a pragmatist, and he has adhered closely to a fix-first-and-blame-later approach during the embarrassing launch of the online insurance exchange.

He also has overseen a concentration of decision-making power in the West Wing rather than dispersing it to the agencies. That means that many of those closest to him — and therefore less likely to be fired — are also those most responsible for many of the problems he has had to cope with.

At a Monday event in the Rose Garden, Obama expressed anger at no one in particular about the flaws in the Web site. He called in a “tech surge” of outside experts to fix problematic coding, while defending Sebelius and other government officials for not anticipating the problems.

“This president is looking for solutions, not scalps,” said Jennifer Palmieri, the White House communications director. “His own focus and his clear direction to all of us is to stay focused on getting implementation right.”

The connection between the power to fire and the willingness to do so has been a loose one for many presidents. Many, like Obama, come to office with little or no executive experience. Even those who have served as governors have often shown awkwardness and poor judgment in deciding when to jettison staff.

Ronald Reagan, a two-term California governor who has come to represent decisiveness in presidential leadership, wobbled when it came to personnel. His firing of Chief of Staff Donald Regan, at the behest of his wife, ended an embarrassing public feud between the onetime banking executive and the first lady. Regan found out from another administration official that he’d been dismissed.

Buck-stops-here Harry S. Truman tolerated, for perhaps too long, open criticism from his showy commanding general in Korea, Douglas MacArthur, before relieving him of his duties for not respecting presidential authority. The public responded harshly.

And George W. Bush, the self-described decider, resisted the urging of senior advisers to fire Defense Secretary Donald H. Rumsfeld as the trajectory of the American post-invasion occupation of Iraq plummeted. Bush waited to push Rumsfeld out long after Iraq had descended into sectarian civil war.

“He resisted in part because he believed in loyalty to the people under him and because he bristled at giving in to the chattering class,” said Peter Baker, the New York Times chief White House correspondent and author of “Days of Fire: Bush and Cheney in the White House.”

In announcing the departure in 2006, Bush said Rumsfeld “understands that Iraq is not working well enough, fast enough.”

Obama used similar words this week to describe the problematic site. But he has appeared reluctant to fire people throughout his administration.

As the BP oil spill stained the Gulf of Mexico for weeks in the spring of 2010, many lawmakers clamored for the dismissal of Interior Secretary Ken Salazar.

Nope, Obama answered.

Then-Secretary of State Hillary Rodham Clinton was a target of firing demands after attacks in Benghazi, Libya, on Sept. 11, 2012, killed the U.S. ambassador and three other Americans.

No, Obama responded again — although some junior State Department officials were let go in the aftermath of the attacks. Republicans complained that the dismissed underlings were scapegoats.

Robert Dallek, a presidential historian, said that if the fired official in the aftermath of a screw-up is “a low-level technocrat, politically it can do a president more harm than good.”

“If it’s a name that has clout with the public — a general, for example — then it can give the president some credibility,” he said. “But if it’s some bureaucrat, most people think, ‘Well, what is this about? Why are they being fired?’ ’’

When Obama has fired senior officials, his reasons for doing so appear to fall into three general categories: obvious incompetence, a change in policy direction, or a mistake by an otherwise admired appointee that makes keeping the job untenable.

An example of the first happened just a few months after Obama took office. A “furious” Obama fired Louis Caldera as head of the White House military office for arranging an unannounced Air Force One flyover of Lower Manhattan, an incident that evoked the Sept. 11, 2001, attacks for many terrified New Yorkers.

An example of the second category — policy change — came a month later, in May 2009. As he began revising Afghanistan war strategy, Obama dumped the commanding general there, Gen. David D. McKiernan, and replaced him with Gen. Stanley A. McChrystal, an advocate of a more refined counterinsurgency approach.

McChrystal became an example of the third category a year later. After he mocked the administration in general and Vice President Biden in particular in a magazine article, Obama summoned him to Washington and fired him. In Obama’s view, McChrystal had shown carelessness and disrespect, making his position untenable.

McChrystal’s successor, Gen. David H. Petraeus, eventually ended his career in the Obama administration much the same way, offering his resignation after an extramarital affair was made public.

While often reluctant to fire, presidents also have resources at their disposal to keep potentially disgruntled former advisers content and out of sight, if necessary.

“There are always some nice ambassadorial posts around,” Hess said, “and you often want to give these folks something. ”