The rollout of the health care law has been plagued with problems so deep that even some of its strongest supporters have soured on its potential. The bottlenecks in the federal online insurance exchange, which serves 36 states; the cancellation of hundreds of thousands of policies that did not comply with the minimum requirements of the new law; and the high price of some plans sold through both federal and state-run exchanges have all cast a pall over President Obama’s efforts to win support for the law.

But for all those problems, people are enrolling. More than 243,000 have signed up for private coverage through the exchanges, according to the Kaiser Family Foundation, and more than 567,000 have been determined eligible for Medicaid since the exchanges opened on Oct. 1. For many, particularly people with existing medical conditions like Mr. Acosta, the coverage is proving less expensive than what they had. Many others are getting health insurance for the first time in years, giving them alternatives to seeking care through free clinics or emergency rooms — or putting it off indefinitely.

Not all who need health insurance are happy with their new options. Many have complained that the prices are too high, especially if they earn too much to qualify for federal subsidies. And many will have a limited choice of doctors and hospitals under the new exchange plans, which have “narrow networks” to hold down premiums.

But Mr. Acosta, like the people in the following profiles, says the health care law has given him a cautious sense of hope.

He lives alone and has turned to his siblings at times for help with his insurance and medical costs. And those costs have at times been high: He sees an oncologist in Tucson twice a year and sometimes needs expensive tests that his current insurance does not always cover. He has also recently been stretching his medication, a drug called Gleevec, which helps make his type of leukemia manageable, to ensure he would have some on hand if he became uninsured.