The new year brought relief for Americans who previously had no health insurance or were stuck in poor plans, but it also led to confusion after the troubled rollout of the federal health care law sent a crush of late applications to overloaded government agencies.

That created stacks of yet-to-be-processed paperwork and thousands — if not millions — of people unsure about whether they have insurance.

Mike Estes of Beaverton, Ore., finally received his insurance card on Dec. 27 after applying in early November. Still, the family was thrilled to have insurance through the Oregon Health Plan, Oregon’s version of Medicaid, because their previous $380-a-month premium “literally crushed our family’s finances,” Estes said.

Dr. John Venetos, a Chicago gastroenterologist, said there was “tremendous uncertainty and anxiety” among patients calling his office.

“They’re not sure if they have coverage. It puts the heavy work on the physician,” Venetos said. “At some point, every practice is going to make a decision about how long can they continue to see these patients for free if they are not getting paid.”

Obama administration officials estimate that 2.1 million consumers have enrolled so far through the federal and state health insurance exchanges that are a central feature of the federal law.

But even before coverage began, health insurance companies complained they were receiving thousands of faulty applications from the government, and some people who thought they had enrolled for coverage have not received confirmation.

Tens of thousands of potential Medicaid recipients in the 36 states relying on the federal exchange also are in limbo after the federal website that was supposed to send their applications to the states failed to do so.

Reports of other complications were scattered around the country.

In Burlington, Vt., the state’s largest hospital had almost two dozen patients seek treatment with new health insurance policies, but more than half of those did not have insurance cards.

Minnesota’s health care exchange said 53,000 people had enrolled for coverage through its marketplace, but it was unable to confirm the insurance status of an additional 19,000 people who created accounts but did not appear to have purchased plans.

In Connecticut, officials were pleading for patience as call centers fielded calls from people who are concerned because they had yet to receive a bill for premiums or an insurance identification card.

“This is an unprecedented time, because there are a record number of people who have applied for coverage with an effective date of Jan. 1,” said Donna Tommelleo, a spokeswoman for the Connecticut Department of Insurance.

But the volume was no higher than usual Thursday at the call center serving the federal health care exchange, where the vast majority of calls were from consumers seeking coverage starting Feb. 1, U.S. Health and Human Services spokeswoman Joanne Peters said.

The agency is coordinating with “insurers, providers, hospitals, and pharmacists to help smooth the transition for consumers who are using their new plans for the first time,” she said in an emailed statement.

The major pieces of the Affordable Care Act that took effect with the new year mean people with pre-existing medical conditions cannot be denied coverage, yearly out-of-pocket medical expenses will be capped and new insurance policies must offer a minimum level of essential benefits, ranging from emergency room treatment to maternity care.

Some parts of the law took effect previously, such as the ability of young people to remain on their parents’ insurance policies until age 26.

The new year brings the most personal test yet for President Barack Obama’s health care overhaul as millions of patients begin to seek care under its new mandates. The burden for implementing the law now shifts to insurance companies and health care providers.

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The entire tracking system was “in a sort of chaos” Thursday as consumers tried to use or confirm their new insurance, said Kelly Fristoe, an insurance agent in Wichita Falls, Texas.

“I’ve got pharmacies that are calling in to verify benefits on these new plans that are getting incorrect information,” he said. “I have people that are calling to make their initial premium payment, and they’ve been on hold for maybe three or four hours at a time and then they get hung up on.”

People who signed up on the federal website have until Jan. 10 to pay premiums for coverage retroactive to Jan. 1, while consumers in some states have until Jan. 6.

Premiums paid after the deadline will be applied to coverage starting Feb. 1 or later. Consumers have until March 31 to sign up in time to avoid a federal tax penalty for remaining uninsured. That fine starts at $95 for an individual this year but climbs rapidly, to a minimum of $695 by 2016. There is an additional fine for parents who do not get health insurance for their children.