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