To be sure, the launch of Obamacare exchanges, which signed up about 11.4 million people this past open enrollment season, coupled with the expansion of Medicaid benefits in the majority of states, has led to a sharp decrease in the uninsured rate over the past two years.
The unsinured rate in the first quarter of 2015 was 11.9 percent, according to the Gallup polling organization. Right before the opening of the Obamacare exchanges in late 2013, the uninsured rate was a record high 18 percent, Gallup has said.
The Obama administration has touted that reduction and repeatedly stressed the fact that federal subsidies are available to most enrollees on the government-run exchanges.
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"I think in the first or second years of the [Affordable Care Act] you picked up the easier people," Hamel said, particularly those with chronic health conditions who felt a strong financial need for insurance. However, for the remaining uninsured, "I do think that it will be harder and harder over time to convince those people to sign up," she said.
Another finding in Kaiser's survey showed how even people who did sign up for individual health plans have concerns that their coverage won't be good enough to completely protect them financially. Nearly six in 10 respondents said they had fears they would not be able to cover the costs for the medical expenses.
Those concerns were particularly acute people in so-called high deductible coverage, or plans in which they could have to pay $1,500 or more in total out-of-pocket-costs per individual for medical bills. About 40 percent of respondents had such plans.
When asked a hypothetical question about an unexpected $1,500 bill not covered by their plan, 31 percent of people in high-deductible plans said they could pay it off without going into debt. But 43 percent said they would either have to go into credit card debt or borrow to cover the bill. Fifteen percent said they would be unable to pay the bill, and the remainder said they would work out a payment plan with their provider.
Hamel pointed out that there has been a trend toward high-deductible plans for several years. As that trend continues, "more and more people are likely to run into this financial problem," she said.
Kaiser's study comes a day after the Commonwealth Fund issued a report on the problem of 31 million Americans being "underinsured," meaning they have insurance but still face potentially high out-of-pocket medical bills relative to their incomes.
The Commonwealth Fund study found that high deductible plans were playing a greater role in making people underinsured. That, in turn, leaves them at bigger risk of going into debt to pay their bills, having to file for bankruptcy, or skipping medical treatment.