In 2014, for example, 35 percent of adults, or 64 million people reported some kind of bill problems or medical debt, compared with 41 percent of the population, or 75 million people in 2012, according to the Commonwealth Fund's survey released Thursday.
Read MoreHealthCare.gov tops 2014 tally
Also in 2014, the number of adults who reported not getting needed care because of cost, including visiting a doctor, filling a prescription or getting specialist care had decreased to 66 million, or 36 percent. In 2012, 80 million people, or 43 percent of the population, reported such issues.
"These declines are remarkable and unprecedented in the survey's more than decade-long history," said Sara Collins, the Commonwealth Fund's vice president for health-care coverage and access and lead author of the study.
"They indicate that the Affordable Care Act is beginning to help people afford the health care they need. We also found sharp declines in the uninsured rate nationwide," Collins said.
The survey suggests there could be continued decreases in the number of people facing health-care financial pressure in coming years, as more people obtain health insurance through elements of the Affordable Care Act, particularly among young and poor adults.
But Commonwealth Fund President Dr. David Blumenthal warned that a Supreme Court case that is attacking one of those major features could undo progress seen in the levels of people strapped by health costs and in number of people who lack insurance.
That case challenges the legality of subsidies given to most customers of the federal Obamacare exchange HealthCare.gov—which sells insurance plans in 37 states—to help them pay for premiums and out-of-pocket health costs.
"I think the Supreme Court decision will have a major effect...if there is a decision in favor of the plaintiffs we would expect, in those states at least in the short term, the situation would revert to the kinds of numbers we saw before the Affordable Care Act," Blumenthal told reporters during a briefing on the survey. A decision in that case is expected in June.
The non-profit Commonwealth Fund, which seeks to promote "high performance health-care system," since 2001 has conducted a biennial survey of health insurance in the United States. The latest survey, which is based on a sample of 6.027 adults ages 19 and older, ran from last July through December.
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That time-fame encompasses in which the government-run Obamacare exchanges had been selling health insurance plans for at least nine months. And it includes a period which more than half of the U.S. states had expanded their Medicaid programs' eligibility to include nearly all poor adults.
Expanded Medicaid eligibility is a key provision of the ACA, but 23 states have yet endorse expansion beyond their currently eligible residents.
"Unfortunately, we're still seeing high uninsured rates in states that haven't expanded Medicaid," said Collins, the study's author.
Despite that, nearly 10 million of Medicaid recipients have been added to the coverage rolls in the past year nationwide.