NEW YORK — Here is the surest way to enjoy the peace of mind that comes with having health insurance: Don’t get sick.

The number of uninsured Americans has fallen by an estimated 15 million since 2013, thanks largely to the Affordable Care Act. But a new survey, the first detailed study of Americans struggling with medical bills, shows that insurance often fails as a safety net. Health plans often require hundreds or thousands of dollars in out-of-pocket payments — sums that can create a cascade of financial troubles for the many households living paycheck to paycheck.

Carrie Cota learned the hard way that insurance does not guarantee financial security. Cota, a 56-year-old travel agent from Rosamond, Calif., learned she had the autoimmune disease lupus in 2007. She ran up thousands of dollars in medical and dental bills and ended up losing her job, and eventually her house.

“I had to move in temporarily with my ex-husband,’’ she said in a recent interview. “I’m staying with him until I can figure out what to do.’’

In the new poll, conducted by The New York Times and the Kaiser Family Foundation, roughly 20 percent of people under age 65 with health insurance nonetheless reported having problems paying their medical bills over the last year. By comparison, 53 percent of people without insurance said the same.

These financial vulnerabilities reflect the high costs of care in the United States, the most expensive place in the world to get sick. They also highlight a substantial shift in the nature of insurance. Since the late 1990s, insurance plans have begun asking their customers to pay an increasingly greater share of their bills out of pocket through rising deductibles and co-payments. The Affordable Care Act, signed by President Obama in 2010, protected many Americans from very high health costs by requiring insurance plans to be more comprehensive, but at the same time it allowed or even encouraged increases in deductibles.

“We’re at a point where there’s been slow growth in health care costs and huge improvements in the numbers of people who have health insurance,’’ said Sara Collins, a vice president at the Commonwealth Fund, a health research group. “But there is this underlying trend toward higher cost-sharing that could put increasing numbers of people at risk for being underinsured.’’

Among those who reported having problems paying their bills despite having insurance, 63 percent said they used up all or most of their savings; 42 percent took on an extra job or more work hours; 14 percent moved or took in roommates; and 11 percent turned to charity.

Randy Farris, 58, a factory worker from Conger, Minn., needed a knee replacement three years ago. His insurance covered 80 percent of the bill, but he needed to cash in an IRA to pay his $4,000 share. “I haven’t been to the doctor since because I don’t want any more doctor bills,’’ he said. His wife’s retirement savings had been wiped out years before, he said, when he used them to pay her hospital bills after she died of cancer.

The health law led to a decline in the number of Americans suffering financial stress from health problems, thanks to the options for receiving coverage, especially for the poor. But the problem is still widespread, touching roughly a quarter of Americans under 65, when the insured and uninsured are looked at together. Americans older than 65 are covered by Medicare, which more frequently protects people from major financial trouble.

Unlike other polls, which have focused on the ways that insurance affects health care, the new Times-Kaiser survey explored the effects of medical bills on people’s daily lives well beyond the medical system. It found that medical bills don’t just keep people from filling prescriptions and scheduling doctors’ visits. They can also prompt deep financial and personal sacrifices, affecting their daily lives.

“The major impact is actually a pocketbook or economic impact: their ability to pay the rent or the mortgage or buy food,’’ said Drew Altman, president of the Kaiser Family Foundation.

People without insurance, of course, are more vulnerable to medical bills than those with coverage. The study found that the people most likely to report bill problems were uninsured, poor, or disabled.