When Roron Chen went to a public clinic to try to see a doctor, she was put on the waiting list at the San Mateo Medical Center. A California state law requires counties to provide health care to people like Ms. Chen who cannot afford health insurance and have no other alternatives.

Ms. Chen, a homemaker from San Mateo whose husband lost his job three years ago, had to wait nearly a year to see a doctor through the program. Like thousands of other uninsured people in San Mateo County, Ms. Chen waited and waited because the county did not have the money to subsidize her health care.

At the time, the tax-supported Peninsula Health Care District held a $43 million reserve of current public assets to further its mission statement, which includes “responding to the local health needs and allocating resources for programs that enhance the health of the community.” The county had asked the district to dip into its reserve to contribute $4 million for subsidized health care for residents like Ms. Chen, but the district refused.

The Peninsula Health Care District is one of about 30 such health care districts, out of 74 statewide, that no longer run hospitals — a departure from their original mission when, after World War II, the State Legislature allowed communities to levy taxes to support hospitals because the state’s rural and low-income areas lacked access to basic hospital care.