A majority of Americans with health insurance said they had encountered obstacles to coverage, including denied medical care, higher bills and a dearth of doctors in their plans, according to a new survey from KFF, a nonprofit health research group. As a result, some people are delayed or skipped treatment.

Those who were most likely to need medical care — people who described themselves as in fair or poor health — reported more trouble; three-fourths of those receiving mental health treatment experienced problems.

“The consequences of care are delayed and missed altogether because of the sheer complexity of the system which is significant, especially for people who are sick,” said Drew Altman, the chief executive of KFF, formerly known as the Kaiser Family Foundation.

The survey also underscored the persistent problem of affordability as people struggled to pay their share of health care costs. About 40 percent of those surveyed said they had been delayed or gone without care in the last year because of the expense. People in fair or poor health were more than twice as likely to report problems with paying medical bills than those in better health, and Black adults were more likely than white adults to indicate they had trouble.

Nearly half of those who encountered a problem with their insurance said they could not satisfactorily resolve it. Some could not obtain the care they had sought, while others said they paid more than expected. Among the nearly 60 percent who reported difficulties with their insurance coverage, 15 percent said their health had declined.

“This survey shows it’s not enough to just get a card in your pocket — the insurance has to work or it’s not exactly coverage,” said Karen Pollitz, the co-director for KFF’s patient and consumer protection program.

People have a hard time understanding their coverage and benefits, with 30 percent or more reporting difficulty figuring out what they will be required to pay for care or what exactly their insurance will cover.

“Insurances are way more complicated than they should be,” said Amanda Parente, a 19-year-old college student in Nashville who is covered under her mother’s employer plan. She was surprised to find that her out-of-pocket costs spiked recently when she sought treatment for strep throat. While she realized her co-payments would be higher, “I guess we didn’t know how drastic it was going to be,” she said.

Navigating the intricacies of coverage and benefits were similar regardless of what kind of insurance people had. At least half of those surveyed with private coverage, through an employer, those with an Obamacare plan, or a government program like Medicare or Medicaid, said they experienced difficulties.