They told me that they were not on the plan, they were never on the plan and they'd been trying to get their name off the plan all year," said Papa, who recently bought a plan from a different insurance company.

It was no better with the next doctor she was assigned. The Naples, Fla., resident said she left a message to make an appointment, "and they never called back."

The Papas were among the 6.7 million people who gained insurance through the Affordable Care Act last year, flooding a primary care system that is struggling to keep up with demand.

A survey this year by The Physicians Foundation found that 81 percent of doctors describe themselves as either over-extended or at full capacity, and 44 percent said they planned to cut back on the number of patients they see, retire, work part-time or close their practice to new patients.

Newly insured struggle to find primary physicians