This story is part of a reporting partnership that includes KUHF, and Kaiser Health News.

At the Martin Luther King Jr. Health Center on Houston’s south side, patients can fill a prescription, see a podiatrist, and get new eyeglasses. They can even buy fresh fruits and vegetables at a weekly farmer’s market. But what they can’t always get is a doctor’s appointment.

“There’s no way possible that the doctors can see any more than they can see,” says Carolyn Bell, the center’s director. “They’re overbooked, overcapacity, overworked. So, what we do — we screen and we treat the most acute.”

Surveys have repeatedly shown that one-third of adults in Harris County, which includes Houston and some of its suburbs, have no health insurance. For most of them, the hospital district is the medical safety net.

The district is partially funded by property taxes, and includes three hospitals, 44 clinics and a dialysis center. Still, it’s not nearly enough to meet all the medical needs of the uninsured.

Dr. Robert Trenschel, who is in charge of the clinics, compares the situation to “pouring 20 ounces of water into a 16-ounce glass. There’s just huge demand, and there’s not enough capacity currently to accommodate that demand.”

For example, every day more than 300 people call, but are unable get an appointment with a primary care doctor. Returning patients also have trouble. Monica Smith sees a doctor regularly for her diabetes and high blood pressure.

“Sometimes you have to wait a long time to get an appointment with him, because he has so many patients,” Smith says of her doctor. “That’s just how it is.”

Until recently, the hospital district was not able to quantify the extent of the problem. But now Trenschel has access to data from the system’s electronic medical records. He found that most doctors in the district have a heavier patient load than medical associations recommend.

Trenschel warns that the situation in Houston is likely to worsen in 2014, when the Affordable Care Act allows 32 million more people nationally to get subsidized health insurance or Medicaid coverage.

“This is just where we are right now,” he said. “As new patients get insurance, as new patients are able to access the system, there’s this concept called pent-up demand so when patients finally get insurance, then the demands on the system really become exponential.”

Trenschel notes that just to meet current demand, the hospital district would need to hire dozens of new doctors, nurses and assistants.