Millions of Americans visited new online health insurance exchanges as enrollment opened on Tuesday, suggesting a broad national appetite for the affordable coverage that President 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. Federal officials said more than 2.8 million people had visited HealthCare.gov, the federally run exchange that serves residents of more than 30 states, though the figure would include those who received error messages. State-run exchanges also reported higher-than-expected use, including several million visits to New York’s Web site.

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

But it remained unclear whether the array of problems — many people received 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 design. Federal and state officials had promised for months that the exchanges would be ready for heavy use by Oct. 1, and had run numerous tests to ensure that the complex systems would work properly from the start.