“While the holiday surprises have become commonplace, the latest extension is another disappointment for the ‘most transparent administration in history,’ ” Representative Fred Upton, Republican of Michigan, the chairman of the House Energy and Commerce Committee, said in a statement. “As we celebrate Christmas and prepare to ring in the new year, we continue to ask, what’s next?”

Mr. Obama, vacationing in Hawaii with his family, offered a symbolic gesture to people who are using the government’s health care site to get insurance: White House officials announced Monday that the president had enrolled in a health insurance plan over the weekend by using the District of Columbia’s website.

The president and his family receive health care through the military. But administration officials said they hoped that Mr. Obama’s gesture would set an example for many of the uninsured they were urging to sign up before the deadline.

Even though Mr. Obama will not actually use his new plan, a White House official said, “he was pleased to participate in a plan as a show of support for these marketplaces, which are providing quality, affordable health care options to more than a million people.”

The president opted for a bronze plan, one of the less expensive options on the HealthCare.gov site.

For many ordinary visitors who had postponed visiting the health care website, Monday was a frustrating day. While it appeared that the site did not crash or go offline, people calling the federal exchange on Monday morning were greeted with a recorded message: “Welcome to the health insurance marketplace. We are currently experiencing longer-than-normal hold time. Do you need coverage for Jan. 1? Please stay on the line, as your call is important to us.”

Callers were told to be prepared for a long wait. “If you are using a cell- or cordless phone, make sure it’s fully charged,” the message said. In the early evening, some were told that they would have to wait 30 minutes. Others heard a recorded message that said: “Due to heavy call volumes, we are unable to take your call at this time.”