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— Stephanie and Daniel Dufek did everything they were supposed to do in building a middle-class life for themselves and their daughter. They lived modestly. They had health insurance. They saved.

“We had a normal life,” Stephanie Dufek said.

Then her husband got sick.

In roughly a year, the family incurred more than $20,000 in medical bills, shattering their finances.

Their experience shows the risks that even families with health insurance can face when confronted by a serious illness. And more families have found themselves in this situation as health plans with high deductibles — and particularly with high caps on out-of-pocket expenses — have proliferated.

Like many families nowadays, the Dufeks had a health plan with a high deductible — $4,000 in their case. But that alone is not what overwhelmed them.

The blow came first from the health plan’s high cap on out-of-pocket expenses, and then from the family’s facing the same expense again with the start of a new year.

That’s a risk that remains even with the new limits on out-of-pocket expenses under the Affordable Care Act. The law caps those expenses for an individual at $6,600 this year. But that means that someone with a health plan that runs on the calendar year and who is diagnosed with a serious illness in October, for example, can easily incur $13,200 in medical bills in just a few months, on top of the premiums they pay for insurance.

The Dufeks’ health plan wasn’t covered by the Affordable Care Act, and their plan had an unusually high limit on out-of-pocket expenses for hospital stays: 10 percent coinsurance up to $10,000 a year.

The result was a total of $14,000 in expenses from the deductible and coinsurance — and the prospect of an additional $14,000 in bills the next year.

Their situation is unusual. But it happens, and when it does, the results can be devastating.

A report by the Consumer Financial Protection Bureau found that medical debts account for 52 percent of the actions by debt collection agencies that appear on consumer credit reports, according to a report by the Kaiser Family Foundation.

The report also showed that many of the people who had been contacted by collection agencies about medical bills previously had no history of credit problems.

Health plans with high deductibles and high caps on out-of-pocket expenses have become widespread. They are a way of lowering premiums, and that benefits most people because they have the good fortune to be healthy. The plans also are thought to have had a part in slowing the rise in health care spending by encouraging people to be better consumers of health care.

But plans with high out-of-pocket expenses also shift risk, particularly to people who need health insurance the most: those who become seriously ill.

“It’s not a perfect allocation of risk,” said Martha Gaines, a lawyer and director of the Center for Patient Partnerships at the University of Wisconsin Law School. “It’s a bad way to take care of people who are chronically sick.”

The costs for some people — such as those who must take high-cost specialty drugs — are incurred year after year.

The Dufeks’ high-deductible health plan had a health savings account, and Dan’s employer contributed $1,000 to the account each year. In some years, the family, who live in Oak Creek, Wisconsin, had money in the account that carried over to the next year.

“Until Dan got sick, it was fine,” Stephanie Dufek said.

Dan Dufek, 43, who is close to 6-foot-5 and 220 pounds, began to have back pain in early 2013. He went to a chiropractor, believing he had a herniated disc. The pain only got worse. Dufek, who delivered restaurant supplies, continued working, telling himself that he would just “push through it.”

“Bills have to be paid,” he said. “I have a daughter.”

In October of that year, he finally relented and went to a physician who specializes in back pain.

“I look back on it now and realize that I was in denial,” Dufek said.

The physician ordered an MRI that showed a large tumor in his spine.

It was later diagnosed as a rare, noncancerous tumor. The surgery to remove it took nine hours and required an orthopedic surgeon and a neurosurgeon. The doctors at the time thought Dan might never walk again.

The family hit the $10,000 limit on coinsurance for hospitalization in the health plan. Within a few months, a new plan year would start, and they again would have a $4,000 deductible and the prospect of paying 10 percent of the cost of any hospital stays.

Those sorts of financially punishing bills undercut the reason people have insurance.

“The most basic reason for insurance is to protect you from those catastrophic costs,” said Thomas Buchmueller, a professor of risk management and insurance at the University of Michigan.

An estimated 4 million people with middle-class incomes — those earning about $47,000 to $95,000 for a family of four — in 2012 were underinsured, according to a report by the Commonwealth Fund last year.

The report defined underinsured people as those who had health insurance but still spent 10 percent or more of their incomes on medical costs.

A torrent of bills followed Dan Dufek’s diagnosis and surgery.

“I can bring you the box,” Stephanie Dufek said.

The cardboard box — a carton that once held boxes of Girl Scout cookies — is where she throws the medical bills.

It is half full.

Some of the bills are incomprehensible, leaving the couple baffled by what is covered by insurance and what they owe. A bill dated Oct. 31, 2014, from Froedtert Health for an MRI, for example, lists charges of $18,499.90, a payment adjustment of $16,959.90 and a patient balance of $1,514.60. This was long after the family had surpassed the deductible on their health plan.

Although the family once had savings, they now are on several payment plans.

After recovering from his surgery, Daniel Dufek, who walks with a cane, went back to his job for a few weeks but soon realized that he could no longer work. He had been prudent and had long-term disability insurance, which now pays about half his previous wage.

Stephanie Dufek, who had worked part time, went back to work full time last April. The job provides health benefits but has a $3,000 deductible. Between their two health plans, the deductibles totaled $7,000 last year. The family went through both of them.

On a recent night, Stephanie Dufek had just written a $60 check to Froedtert Health — money that she said could buy shoes for her daughter or groceries or a nice dinner out.

That is in the past.

They no longer are putting money in their daughter’s college fund. They recently had to buy a new furnace. They keep the thermostat low. And their tax refund is already spent.

“You get your taxes done,” Dan Dufek said, “and it’s going to go to medical bills.”

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