WASHINGTON >> A flood of last-minute applicants rushed to sign up for health insurance on Monday, deadline day for President Barack Obama’s health care law, with more than 100,000 people at a time using the fragile system despite a new spate of intermittent ills.

Supporters of the overhaul fanned out across the country in a final dash to sign up uninsured Americans.

The HealthCare.gov website stumbled early on Monday — out of service for nearly four hours as technicians patched a software bug. The system came back up shortly before 9 a.m. EDT, then another hiccup in early afternoon temporarily kept new applicants from signing up. The system, overwhelmed by computer problems when launched last fall, has been working much better in recent months and officials said it was operating at full capacity on Monday.

At Chicago’s Norwegian American Hospital, people began lining up shortly after 7 a.m. to get help signing up for subsidized private health insurance.

Lucy Martinez, an unemployed single mother of two boys, said she’d previously tried to enroll at a clinic in another part of the city but there was always a problem. She’d wait and wait and they wouldn’t call her name, or they would ask her for paperwork that she was told earlier she didn’t need, she said. Her diabetic mother would start sweating so they’d have to leave.

She’s heard “that this would be better here,” said Martinez, adding that her mother successfully signed up Sunday at a different location.

“Organizations across the country have been mobilizing for this moment,” said Ron Pollack, executive director of Families USA, an advocacy group that has supported the health care overhaul from its inception. “I think we are going to see huge numbers of people get involved.”

At a Houston community center, there were immigrants from Ethiopia, Nepal, Eritrea, Somalia, Iraq, Iran and other conflict-torn areas, many of them trying anew after failing to complete applications previously. In addition to needing help with the actual enrollment, they needed to wait for interpreters. Many had taken a day off from work, hoping to meet the deadline.

The White House and other supporters of the law were hoping for an enrollment surge that would push sign-ups in the new health insurance markets to around 6.5 million people. That’s halfway between a revised goal of 6 million and the original target of 7 million. The first goal was scaled back after the federal website’s disastrous launch last fall, which kept it offline during most of October.

The insurance markets — or exchanges — offer subsidized private health insurance to people who don’t have access to coverage through their jobs. The federal government is taking the lead in 36 states, while 14 other states plus Washington, D.C., are running their own enrollment websites.

Of the 6 million people nationally who had signed up before the weekend, it’s unclear how many ultimately closed the deal by paying their first month’s premium. Also unknown is how many were previously uninsured — the real test of Obama’s health care overhaul. In addition, the law expands coverage for low-income people through Medicaid, but only about half the states have agreed to implement that option.

Cheering on the deadline-day sign-up effort, Health and Human Services Secretary Kathleen Sebelius planned to spend much of the day Monday working out of the department’s TV studio, conducting interviews by satellite with stations around the country.

Though March 31 was the last day officially to sign up, millions of people are potentially eligible for extensions granted by the administration.

Those include people who had begun enrolling by the deadline but didn’t finish, perhaps because of errors, missing information or website glitches. The government says it will accept paper applications until April 7 and take as much time as necessary to handle unfinished cases on HealthCare.gov. Rules may vary in states running their own insurance marketplaces.

The administration is also offering special extensions to make up for all sorts of problems that might have kept people from getting enrolled on time: Natural disasters. Domestic abuse. Website malfunctions. Errors by insurance companies. Mistakes by application counselors.

To seek a special enrollment period, contact the federal call center, at 1-855-889-4325, or the state marketplace and explain what happened. It’s on the honor system. If the extension is approved, that brings another 60 days to enroll.

Those who still don’t get health insurance run the risk that the Internal Revenue Service will fine them next year for remaining uninsured. It remains to be seen how aggressively the penalties called for in the law are enforced.

Also, the new markets don’t have a monopoly on health insurance. People not already covered by an employer or a government program can comply with the insurance mandate by buying a policy directly from an insurer. They’ll just have to pay the full premium themselves, although in a few states there may be an exception to that rule as well.

Supporters of the law held their breath early Monday when the website was taken down.

Visitors to HealthCare.gov saw messages that the site was offline for maintenance. At times the visitors were also directed to a virtual waiting room — a feature designed to ease the strain on the site during periods of heavy use.

Administration spokesman Aaron Albright said the site undergoes “regular nightly maintenance” during off-peak hours and the period was extended because of a “technical problem.” He did not say what the problem was, but a statement from the Department of Health and Human Services called it “a software bug” unrelated to application volume.

The site, which was receiving 1.5 million visitors a day last week, received about 1.7 million on Sunday.

A recent analysis for The Associated Press by the performance-measurement firm Compuware found that the government site runs slowly compared with health insurance industry peers.

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Associated Press writers Connie Cass in Washington, Don Babwin in Chicago and Ramit Plushnick-Masti in Houston contributed to this report.