Abby Goodnough, Robert Pear and Richard Pérez-Peña / New York Times News Service

Millions of Americans visited new online health insurance exchanges as enrollment opened Tuesday, suggesting a broad national appetite for the affordable coverage that President Barack Obama has promised with his health care law. But many people quickly encountered technological problems that prevented them from getting rates, comparing health plans or signing up.

Federal and state officials said that while they knew there was pent-up demand for health coverage, the number of visits to their exchanges was greater than anticipated. By late Tuesday, federal officials said more than 2.8 million people had visited, the federally run exchange that serves residents in more than 30 states. State-run exchanges also reported higher-than-expected use, including several million visits to New York’s website.

The demand “exceeds anything that we had expected,” Obama said at the White House on Tuesday afternoon.

The number of visits to the federal exchanges was seven times as high as had ever been experienced at, federal officials said, although it would include people who received error messages.

But it remained unclear whether the array of problems — many people received error messages saying the system was down and others were unable to create accounts to buy insurance — stemmed more from heavy traffic or from flaws in how the exchanges were designed. Federal and state officials had promised for months that the exchanges would be ready for heavy use by Tuesday and had run numerous tests to ensure that the complex systems would work properly.

The problems came on a landmark day for the health care law, Obama’s signature legislative achievement and the central issue in the congressional battle over federal spending that has led to a partial shutdown of the government. A trouble-plagued start, accompanied by the complaints of frustrated consumers, could undermine support for the law and discourage people from signing up.

Across the country, people trying to use the Internet exchanges expressed mixtures of hope and frustration.

At the Jessie Trice Community Health Center in Miami, out of about 70 people who arrived seeking help with the new program before noon, no one was able to get access to the marketplace. All were given appointments to come back another week.

At the Lower Light Christian Health Center, a clinic just west of downtown Columbus, Ohio, Carrie Moss, a certified application counselor, said the exchange website was not working when she tried to show patients their options.

And in Kentucky, by 9 a.m. the state-run insurance exchange was freezing up, with users unable to take even the first step of creating an account.

Gwenda Bond, a spokeswoman, said that more than 1,000 applications had been filed online by 8:30 a.m. and that the “high volume of traffic” was to blame for the problems.

“This surge of early applications demonstrates the pent-up demand for quality health coverage,” Bond said.

By late afternoon in the East, state exchanges reported much heavier traffic to their websites than expected. New York said that 10 million attempts had been made to reach its site- although with many people making multiple tries, it was not clear how many individuals that represented. Officials said the figure was so far beyond anything they had considered plausible that they were investigating the cause.

In Illinois, the exchange site had 70,000 visits from 65,000 people, with 1,100 applications submitted. Kentucky had 58,000 unique visitors, who started nearly 2,000 applications and completed more than 1,200. Oregon’s exchange had 68,000 visitors, but with the state still working out some problems, it had postponed the ability to apply for coverage — possibly for weeks.

About 55,000 people went to Colorado’s exchange site, and 1,450 had created accounts to allow them to start shopping for plans. Idaho’s exchange had more than 8,000 visitors, but it was unclear whether any of them had been able to apply, because the applications were routed through the federal system.

In California, Peter Lee, the executive director of Covered California, said the state health exchange was running slowly because it had received 5 million hits Tuesday, and he said workers were performing maintenance to speed it up.

“Some of the folks have had a slow experience,” Lee said. “Surprise, surprise.”

Critics quickly seized on the technology problems as evidence that the health care law was deeply flawed.

“We should have never gotten to this point,” said Sen. Orrin Hatch of Utah, the senior Republican on the Senate Finance Committee. “The Obama administration should have acknowledged the ample warning signs of problems in the exchanges and heeded the many calls for delay.”

But in his Rose Garden appearance, Obama staunchly defended the law, saying “This is life-or-death stuff.”

“Tens of thousands of Americans die each year just because they don’t have health insurance,” the president said. “Millions more live with the fear that they’ll go broke if they get sick. And today, we begin to free millions of our fellow Americans from that fear.”

In all, the Obama administration expects about 7 million people to be enrolled in private health plans through the exchanges in 2014. Millions more are expected to get coverage under an expansion of Medicaid, although many states are refusing to go along with the expansion because their leaders are opposed to the health care law.