NASHVILLE — Two nights a year, Tennessee holds a health care lottery of sorts, giving the medically desperate a chance to get help.

Adrian Casteel, 55, of Nashville, has a heart condition and a steel plate in his back. He has been on Medicare for nine years, but about 20 percent of his expenses are not covered.

State residents who have high medical bills but would not normally qualify for Medicaid, the government health care program for the poor, can call a state phone line and request an application. But the window is tight — the line shuts down after 2,500 calls, typically within an hour — and the demand is so high that it is difficult to get through.

There are other hurdles, too. Applicants have to be elderly, blind, disabled or the “caretaker relative” of a child who qualifies for Medicaid, known here as TennCare. Their medical debt has to be high enough that if they paid it, their income would fall below a certain threshold. Not many people end up qualifying, but that does not stop thousands from trying.

“It’s like the Oklahoma land rush for an hour,” said Russell Overby, a lawyer with the Legal Aid Society in Nashville. “We encourage people to use multiple phones and to dial and dial and dial.”

The phone line opened at 6 p.m. on Thursday for the first time in six months. At 5:58, Ida Gordon of Nashville picked up her cordless phone and started dialing. Ms. Gordon, 63, had qualified for TennCare until her grandson, who had been in her custody, graduated from high school last spring. Now she is uninsured, with crippling arthritis and a few recent trips to the emergency room haunting her.

“I don’t ask for that much,” Ms. Gordon said as she got her first busy signal, hanging up and fruitlessly trying again, and then again. “I just want some insurance.”

Gov. Bill Haslam, a Republican, has indicated that he will decide this week whether to support an expansion of Medicaid to cover more low-income adults, as called for in the federal health care law. Doing so would add more than 180,000 people to the TennCare rolls by 2019, according to the state, most of them adults like Ms. Gordon whose incomes are within 138 percent of the federal poverty level.

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