“Healthier people chose to keep their plans,” said Amy L. Bowen, a spokeswoman for the Geisinger Health Plan in Pennsylvania, and people buying insurance on the exchange were therefore sicker than expected. Geisinger, often praised as a national model of coordinated care, has requested an increase of 40 percent in rates for its health maintenance organization.

Insurers with decades of experience and brand-new plans underestimated claims costs.

“Our enrollees generated 24 percent more claims than we thought they would when we set our 2014 rates,” said Nathan T. Johns, the chief financial officer of Arches Health Plan, which covers about one-fourth of the people who bought insurance through the federal exchange in Utah. As a result, the company said, it collected premiums of $39.7 million and had claims of $56.3 million in 2014. It has requested rate increases averaging 45 percent for 2016.

The rate requests are the first to reflect a full year of experience with the new insurance exchanges and federal standards that require insurers to accept all applicants, without charging higher prices because of a person’s illness or disability. The 2010 health law established the rate review process, requiring insurance companies to disclose and justify large proposed increases. Under federal rules, increases of 10 percent or more are subject to review.

Federal officials have often highlighted a provision of the Affordable Care Act that caps insurers’ profits and requires them to spend at least 80 percent of premiums on medical care and related activities. “Because of the Affordable Care Act,” Mr. Obama told supporters in 2013, “insurance companies have to spend at least 80 percent of every dollar that you pay in premiums on your health care — not on overhead, not on profits, but on you.”

In financial statements filed with the government in the last two months, some insurers said that their claims payments totaled not just 80 percent, but more than 100 percent of premiums. And that, they said, is unsustainable.

At Blue Cross and Blue Shield of Minnesota, for example, the ratio of claims paid to premium revenues was more than 115 percent, and the company said it lost more than $135 million on its individual insurance business in 2014. “Based on first-quarter results,” it said, “the year-end deficit for 2015 individual business is expected to be significantly higher.”

BlueCross BlueShield of Tennessee, the largest insurer in the state’s individual market, said its proposed increase of 36 percent could affect more than 209,000 consumers.