In a cramped office building in Washington, sandwiched between a deli and a day care, a little-known appeals court for veterans is struggling to work its way through hundreds of backlogged cases.

The court, created two decades ago to weigh disability claims rejected by the Department of Veterans Affairs, is by many measures the nation’s busiest federal appellate court — and a symbol of the accumulating toll of prolonged war, both on soldiers and the bureaucracies created to serve them.

The caseload at the U.S. Court of Appeals for Veterans Claims has doubled in recent years, with the court deciding more than 600 cases per judge each year — far more than other federal appellate courts. Judges are working nights and weekends but say they still have difficulty keeping pace.

Veterans whose claims had already spent years in the VA system often wait several more years for the court to rule on whether they will receive disability payments and free health care. Some have abandoned their appeals. Others, including soldiers from as far back as World War II, have died before a decision was issued.

One veteran’s case lasted 14 years, seven at the appellate court, which considered three appeals in a repeating cycle lawyers dub “the hamster wheel.”

At least part of the blame, say advocates for the nation’s 23 million veterans, rests with the Obama administration, which has made veterans a priority yet has not submitted nominations to fill the court’s bench, where three of the nine seats are vacant.

“I’m hugely disappointed,’’ said Glenn R. Bergmann, president of the bar association that represents veterans before the court. He is one of a number of advocates who have written the White House about the problem, with no response. “It’s not like we’re trying to put someone on the Supreme Court,” he said.

Kate Bedingfield, a White House spokeswoman, said that the administration is working to submit nominations for Senate confirmation as soon as possible and that President Obama “appreciates the critical work” that the court “does on behalf of America’s veterans.”

Legal experts also have blamed the high number of vacancies in the regular federal courts on delays at the White House and in Congress.

At the “veterans court,” the number of cases is surging for several reasons. As veterans from previous wars age, they are facing economic uncertainty and service-related health problems, prompting new claims. With no statute of limitations, the current caseload is mostly appeals from peacetime veterans and those who served in the first Persian Gulf War, Vietnam and, in some instances, Korea and World War II.

Still, the crisis at the court is likely to get worse, lawyers and judges say, pointing to a looming wave of new appeals expected from the VA, where injured Iraq and Afghanistan veterans are filing claims at historically high rates. New claims at the VA have nearly doubled since 2005, and of the 1.3 million living combat veterans discharged since 2001, nearly half have filed for benefits.

In contrast, about 16 percent of Vietnam veterans are receiving benefits.

“The court will become overwhelmed if we don’t get the relief we need,’’ said William Greene, who is among six retired judges — one in his 80s — on whom the court is relying for help. “Having judges with cases in the triple digits is very stressful.’’

Anticipating the surge, Congress in 2008 added two judgeships to what had been a seven-member court, both of which could have been filled 15 months ago.

Although veterans are entitled to a range of benefits, most appeals stem from denials of claims for survivor benefits or disabilities that veterans must prove are service-related.

Numerous Iraq and Afghanistan veterans, facing multiple deployments and surviving wounds that would have killed earlier soldiers, are returning with psychological trauma and brain injuries caused by roadside bombs and other explosive devices.

“In World War II or Vietnam, if you came under mortar attack and you didn’t get killed, you dusted yourself off and went on your way,” said Richard Cohen, executive director of the National Organization of Veterans’ Advocates. “Now, day after day these explosions miss you but knock you around, and soon, you’re having trouble understanding what your squad leader is saying.”

For Daniel W. Frederick, the quest for benefits was prompted by headaches, which began after he contracted meningitis in 1971 while serving at a Naval air station in Texas, according to his attorney, Michael R. Viterna.

He was denied benefits in 1981, reopened the claim in 2003 and was told his VA medical records had been lost.

After the VA rejected his claim in 2009, Frederick appealed to the veterans court, which kept the case an additional 16 months. “It’s terrible it took that long,” said his wife, Margaret.

In February, without holding a hearing, a veterans court judge sent the case back to the VA, saying Frederick was “free to submit additional evidence.”

He had died two months earlier. His widow is now pursuing his benefits.

‘We’re kind of invisible’

No sign announces the U.S. Court of Appeals for Veterans Claims near Judiciary Square. There is none of the grandeur of most courthouses. In fact, there is no courthouse. The court operates out of the top three floors of an office building, with a sole, small courtroom.

The jurists are as anonymous as their location. “The first question we have to answer anywhere we go is: ‘Are you real federal appeals court judges?’ ” said Judge William A. Moorman, a former top Air Force lawyer. “Because we are so specialized, until you deal with us, we’re kind of invisible.”

Congress created the court in 1988, after lobbying by Vietnam veterans. Before that, veterans who were denied benefits had no judicial recourse.

After the caseload began rising in 2005, judges took steps to expedite operations, including a requirement that, in most cases, the veterans and the VA consider a settlement. Nearly all of those cases are sent back to the VA. Lawyers say this leads to years of repeated appeals, while judges say it allows veterans to gather more evidence.

By the time Bruce E. Kasold became chief judge in August, hundreds of cases were awaiting decisions. Retired judges had been helping with motions and simpler cases, but Kasold decided to turn more complex cases over to the retirees, who work with the help of staff attorneys.

“I’m looking at this, and I thought, ‘Whoa, all these cases are sitting in chambers,’ ” said Kasold, a retired Army lieutenant colonel. He said the backlog in chambers has eased, but more than 800 cases are awaiting action earlier in the pipeline, where staff attorneys write memos offering recommendations.

Some lawyers argue that the press of business has forced the court to take shortcuts, a criticism the judges deny.

One senior judge decided two cases, which had been dormant for months, in a single day, according to a lawyer for those veterans. “How can that happen in 24 hours?” said the lawyer, who spoke on condition of anonymity to not offend the court. “Did he even read the briefs?”

‘I felt so horribly betrayed’

The decisions came slower for Philip Cushman, creating a case that stretched over more than three decades.

When he served in Vietnam, a sandbag fell on his back, causing injuries that led to four surgeries. The VA ruled that he was 60 percent disabled, but he said the pain would not allow him to work.

Cushman filed for total disability in 1977 and was repeatedly rejected before winning in 1994. Three years later, when he sought an adjustment that would increase his benefits, he discovered that his medical record had been altered to make his injury appear less severe, according to court documents.

“I felt so horribly betrayed,” said Cushman, 67, of Oregon.

The VA admitted that the record had been altered, court records show, but the agency rejected his new request. Then, Cushman began a seven-year odyssey at the veterans court, which considered three appeals and upheld the VA’s decision.

Cushman then tried the regular federal courts and in 2009, a federal appeals court in Washington ruled that his constitutional rights had been violated.

This year, the VA awarded Cushman nearly $175,000 — payments for benefits he should have received earlier.

Cushman’s attorney, Gordon Erspamer, faulted the VA and the veterans court, saying the court “very assiduously avoided tackling the issue” of his client’s constitutional rights. The judges, he said, are “under huge pressure to find an easy way to dispose of cases.’’

The court declined to comment.