NEW YORK – The rollout of the Affordable Care Act is leading to the cancellation of hundreds of thousands of health insurance policies nationwide, contradicting President Obama’s repeated pledge that people who like their coverage can keep it.

The cancellation notices started to arrive in recent weeks, compounding the political headaches for the White House from the troubled start of its health insurance exchange, the federal website created to give millions of people access to new plans by Jan. 1.

The cancellations come as a result of the Affordable Care Act, enacted in 2010 and being phased in, which says that health insurance policies that do not offer added benefits, such as prescription drug coverage and free preventive care, can’t be sold after this year even if they’re cheaper.

With the federal website expected to face difficulties through November, Americans may have only weeks to find replacement coverage, and many may end up paying higher premiums.

“I keep playing that over and over in my head: that you can keep your health plan, period,” Terri Flay, of Manassas, Va., whose policy is being canceled, said in referring to Obama’s pledge. “But it isn’t ‘period.’ They put a gun to my head saying that I have to pay more because I need the health care insurance.”

The health care law, also known as “Obamacare,” eliminates “substandard policies that don’t provide minimum services,” said Jay Carney, a White House spokesman, in response to the cancellations. The “80-plus percent” of Americans with employer plans or covered by government programs are unaffected.

Obama’s oft-repeated pledge was a central selling point of his health care overhaul, aimed at calming consumers who feared being forced to give up policies and doctors they liked as the plan expanded to many of the nation’s 48 million uninsured.

While it’s unclear how many consumers face cancellation of their insurance nationally, some individual carriers have released data. Florida’s BlueCross BlueShield, for instance, said that about 300,000 members are affected, while California’s BlueShield and Oakland-based Kaiser Permanente will withdraw policies for a combined 280,000.

As many as 80 percent of people who don’t have a company-sponsored plan or insurance through the federal Medicare or Medicaid programs may have to find new health coverage, said Robert Laszewski, an insurance industry consultant in Arlington, Va. About 19 million people are in this market.

Plans bought before March 23, 2010, when the health care law was enacted, can stay in effect under a grandfather clause.

Federal officials should have worked more closely with insurers to better manage the long-coming shift to new coverage, said Erik Gordon, a professor at the University of Michigan’s Ross School of Business.

Instead, “the first thing you get that affects you personally is that you’ve lost your health insurance,” Gordon said in a telephone interview. “That approach is going to backfire politically.”

Republicans seized on the issue, citing Obama’s statements that no one would lose their current plan. Probably no pledge made to sell the bill was more disingenuous than this one,” Brendan Buck, a spokesman for House Speaker John A. Boehner, R-Ohio, said in an email. “But it’s more than just a broken promise; it means a significantly higher health insurance bill for far too many.”

Amid the rush to open the exchange on time, the consumer “was probably in last place” on the priority list for the Obama administration, said Marc Pierce, founder of Chicago’s Stonegate Advisors, a consultant to insurers.

Not all consumers will be losers under the new system. Barbara Wynkoop, 62, of Trexlertown, Pa., said she expects to find a cheaper plan on the exchange, thanks to subsidies and new coverage for her cholesterol medication. She pays $278 a month now for a policy that Capital BlueCross of Harrisburg plans to discontinue.

“The chance for me to save money and still have decent coverage, what my basic needs would be, I’m happy about that,” she said by phone.

Marilyn B. Tavenner, the official most responsible for the rollout of the federal health insurance exchange, blamed outside contractors for the website woes, not her staff, in testimony before a House committee Tuesday. The Centers for Medicare and Medicaid Services, led by Tavenner, was responsible for building and running the exchange website that is supposed to let people compare and buy health plans, aided by tax credits.

Tavenner apologized to Americans, saying the exchange’s flaws are “not acceptable” and vowed that the site “can and would” be fixed. “We are seeing improvement each week,” she told the House Ways and Means Committee. The agency relied on “private-sector contractors, just as it does to administer aspects of Medicare. Unfortunately, a subset of those contracts for have not met expectations.”

Tavenner also said it’s not the Obama administration but insurers who are responsible for the cancellation letters now reaching many people who buy individual policies. And, officials said, people who get cancellation notices will be able to find better replacement plans, in some cases for lower cost.

The policyholders who’ve been getting cancellation notices have been complaining to lawmakers. “Based on what little information the administration has disclosed, it turns out that more people have received cancellation notices for their health care plans this month than have enrolled” in the health care website, said House Ways and Means Committee Chairman Dave Camp, R-Mich., citing a news report of 146,000 cancellations in his state alone.

Other members of the committee chimed in with stories of constituents who had received similar notices. Republicans also offered examples of people being asked to pay more.

Democrats countered by citing constituents who had been able to find lower-cost coverage for 2014 than they have now, some saving hundreds of dollars a month. Rep. Sander Levin of Michigan, ranking Democrat on the panel, said one of his constituents has been paying $800 a month for a BlueCross BlueShield plan and managed to find comparable coverage for $77, after tax credits that lower the premiums.

Still, Levin added, “this has become a matter of legitimate discussion.”

The Associated Press contributed to this report.