Fourteen percent of sick adults in Massachusetts were unable to get health care they needed in the past year, according to a poll conducted by the Harvard School of Public Health and others.

Among them, more than 7 in 10 cited financial reasons. They couldn’t afford the out-of-pocket costs or their insurer refused to cover the test or treatment.

“In a state that prides itself on universal coverage, it’s a finding you wouldn’t expect to see,” said Harvard’s Robert J. Blendon, health policy professor and director of the Harvard Opinion Research Program.

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The poll, released Monday and sponsored by WBUR, the Blue Cross Blue Shield of Massachusetts Foundation, and the Robert Wood Johnson Foundation, found that most participants were satisfied with the quality of health care in the state but that costs increasingly strained their family budgets.

About 98 percent of Massachusetts residents have health insurance, since coverage was expanded under a 2006 law. House and Senate leaders are now considering competing plans for controlling the growth of health costs to bring it in line with or below the estimated rate of growth of the overall state economy.

The poll included 500 people, 18 or older and interviewed in April and May, who said they had been hospitalized or had a serious illness or injury requiring medical treatment in the previous 12 months. Seventy-eight percent of participants said health costs were at least a somewhat serious problem for the state, and 63 percent said the problem has gotten worse over the past five years. Forty percent said they have struggled to pay out-of-pocket costs themselves.

Especially as high-deductible plans become more popular and provider networks shrink, consumers could increasingly find that cost is a barrier to getting care, said Dr. Paul Hattis, co-chair of the health policy task force for the consumer group Greater Boston Interfaith Organization.

“Even if you put an insurance card in everybody's pocket, don’t believe that that solved the access problem,” Hattis said.

A poll conducted last year by Blendon for The Boston Globe showed broad support for the 2006 law, even though 30 percent of people said it was a factor in rising costs.

In the latest poll, drug costs, insurers, and hospitals—in that order—were most often cited by survey participants as major drivers of health costs. Less weight was given to factors that often play prominently in the state debate, including patients getting more treatment than they need, too little government regulation, malpractice issues, and a lack or transparency on costs.

Evidence suggests that drug costs have contributed little to overall increases in health costs in recent years. And while insurers are a factor, about 90 percent of every premium dollar in Massachusetts is required by the state to go toward paying hospitals and doctors for medical care.

Consumers have a hard time distinguishing between the big economic drivers and what they are paying for out of pocket. “The two are not always the same thing,” said Hattis, whose group has been working to educate its member congregations about the complexities of health economics with a series of webinars.

Blendon said that may be problematic for politicians. The public typically is more receptive to a difficult policy change when it provides a solution to a problem as they see it.

Especially as consumers are asked to take a bigger role in making decisions about their health care, “we need to do a better job supporting consumer understanding on how it works,” said Sarah Iselin, president of the Blue Cross Blue Shield Foundation.

Fifty-two percent of participants in the poll said the quality of care in Massachusetts was not a problem, and 75 percent said quality has remained the same or improved over five years. See the poll results at bluecrossmafoundation.org.