This post has been updated to clarify the survey’s findings.

How is this possible? More than one-third of people who are still uninsured under Obamacare — but appear to be eligible for coverage and sought it out — were told that they were not eligible for health insurance under the law, according to a new report.

The finding, from a survey of 10,500 uninsured Americans by the non-profit Kaiser Family Foundation, is truly perplexing. If you extrapolate out to the overall uninsured population in the United States, it means that more than two million people who by the letter of the law should be covered were told that they weren’t qualified.

“I’ve got no possible explanation for you,” Caroline Pearson, vice president of Avalere Health, an independent consulting firm and a top expert on health policy, told TPM. That’s a bit of an exaggeration — she and others had some educated guesses — but it’s not that far off.

Kaiser found that about 48 percent of the estimated 30 million people who remain uninsured were likely eligible for either Medicaid or private insurance subsidies under Obamacare. The rest were ineligible because they made too much money to qualify for tax subsidies or had access to employer-sponsored insurance (17 percent), their immigration status disqualified them (14 percent) or they fell into the law’s coverage gap because their state didn’t expand Medicaid (18 percent).

Of the 48 percent who are eligible for coverage, 41 percent sought it. And of those, 30 percent said that it was still too expensive. Another 33 percent gave another reason, which covers a range of possibilities, including ideological opposition to the law or they didn’t finish their application. But the maddening finding is the 37 percent of eligible uninsured who sought coverage but said they were nonetheless told they didn’t qualify for coverage. The survey didn’t follow up with a question about how they were told they were ineligible.

“It jumped out at us, too,” Larry Levitt, vice president at Kaiser, told TPM in an email.

Two possible explanations came to mind for Pearson and Levitt. One is that some of those people who are eligible for coverage now actually were ineligible when they first tried to apply. The lower a person’s income, the more volatile their income often is. It’s plausible that a chunk of these people had become eligible between the time they applied and the survey, conducted from Sept. 2 to Dec. 15, 2014.

The other is that people misunderstood what they’d been told, whether by the insurance exchange websites, an assistant who helped them try to sign up, etc. The insurance information gap has always been seen as a major challenge to Obamacare.

“People may have been confused about what they were being told,” Levitt said. “Or, they may have been told they were ineligible for marketplace coverage but eligible for Medicaid, or vice versa, and were confused by that or didn’t follow up.”

Some of that responsibility could also lay with HealthCare.gov and the other insurance exchanges. “I think some of the messages for people who appeared to be Medicaid eligible were particularly muddled” in the early days of 2014 enrollment, Pearson said. TPM reached out to administration officials for their take on the Kaiser findings.

But whatever the explanation, it demonstrates a remaining obstacle for Obamacare: Most of the easily enrolled have been swept up in the law’s first two years. Getting through to the rest of the uninsured is the next assignment as the law works toward its goal of near-universal health coverage.