But opening the door to private care - as the embattled VA is in the process of doing for hundreds of thousands of veterans - may not fix the problem, experts say.
Veterans could be entering an overwhelmed system with its own delays and little room left to give. It could also, they say, cause longer delays for those with poor health insurance who already struggle to get care.
"When you've got a system that's at capacity or darn near it - and I would argue we're darn near it - every 100 or 1,000 patients we add, that's 100 or 1,000 others we might not be able to get to," said Travis Singleton, senior vice president at Merritt Hawkins, a physician-recruiting firm that has studied private-sector waits.
Legislation signed last week will soon give veterans who live far from a VA hospital or face long delays the option of receiving federally funded care from a private physician.
In the meantime, the VA has ramped up referrals to private doctors as a way to abate long delays. VA Secretary Robert McDonald on Wednesday told veterans at the AMVETS national convention in Memphis that the agency in the last two months made more than 838,000 referrals outside the VA, 166,000 over the same time last year.
Since each referral results in multiple appointments, the VA has sent about 1.1 million appointments to private doctors in two months alone, McDonald told the group.
Veterans in the audience applauded the news.
It's unclear how many eligible for private care under the new legislation will utilize it. The VA says it is studying the issue.
But directing VA patients to private doctors will stress a system that for years has faced a shortage of doctors and is still measuring the impact of millions of patients newly insured under the Affordable Care Act, according to Richard "Buz" Cooper, a senior fellow at the Leonard Davis Institute of Health Economics at the University of Pennsylvania.
"It's not like there's a big capacity out there looking around for more patients," said Cooper, who has studied physician workforce issues for 20 years.
He said veterans would not be the addition that breaks the system, but doctors who have already increased their workload will have to make more concessions.
That could mean some will turn away patients with Medicaid and Medicare, said Singleton, noting Merritt Hawkins research that shows more doctors are already doing so. He said the problem could get worse over time if doctors prioritize new veteran patients whose care is being paid for by the government.
"To think we have the capacity to do this is foolish," he said. "We're not even meeting demand now."
Increased demand in Philadelphia means a new patient waiting for a primary-care appointment in the private sector waits on average 21 days, according to the Merritt Hawkins study released in January.
The wait for a similar appointment at the city's VA was 34 days, according to data from mid-June.
The wait for a specialty-care appointment at the VA - an average of more than a dozen areas including neurology, dermatology, and gynecology - is 60 days.
That's longer than all of the specialties studied by Merritt Hawkins: five days for orthopedic surgery, six days for cardiology, 22 days for OB/GYN, and 49 days for dermatology.
A spokeswoman from the city's VA hospital said specialists there were more scarce than primary-care providers, and the hospital has recently hired several additional specialists and is recruiting more.
Once patients are established at the VA, the wait times are far better, ranging from four to 11 days on average. But overall, substantial numbers of veterans there are still waiting for care 30 days or longer - the threshold that will make them eligible for private care under the new legislation.
The most recent statistics show more than 8,000 appointments scheduled at the Philadelphia VA - about 13 percent of those on the books at that time - didn't meet the 30-day mark. More than 2,300 took between 61 and 90 days.
Nationally, more than 636,373 appointments scheduled at the VA in mid-July had taken more than 30 days, according to VA statistics.
Some doctors don't share concerns about those patients entering the private sector. Mark Victor, who heads the 30-office Cardiology Consultants of Philadelphia, said his team could take on veterans who face similar health concerns to the patients they see daily "without a blink."
Wojciechowski said he had faced long delays at the VA for care in specialties other than the sleep clinic - including a two-month wait for an appointment with his primary-care doctor.
Still, he said he wanted to hear more about how the private care system will work before deciding if he will use it. But at first blush, he said, he would welcome shorter waits in the private sector.
"Especially for something that could be life-threatening," he said.
Joe Buckley, a disabled Vietnam veteran who works for the VFW, said he wasn't interested in seeking private care, despite often being frustrated with delays at the VA hospital, which he visits every few weeks.
He is concerned the new option will be executed just as poorly as the current VA system.
"I'm totally against it," he said. "I think they should fix what they have."
The Doctor Will See You Now
Average wait time for new patients at the Philadelphia Veterans Affairs hospital:
Average wait times for new patients in the private sector:
Cardiology: 6 days
OB/GYN: 22 days
Orthopedic surgery: 5 days