On Valentine's Day 2009, Scott Crawford, 41 years old, received the break that he thought would save his life. A surgeon at Johns Hopkins Hospital in Baltimore removed his ailing heart and put in a healthy one. The transplant was a success.

But complications put the former tire-warehouse worker in intensive care for almost a year. Surgeons removed his gall bladder, his left leg and part of a lung. And Mr. Crawford soon became one of the most expensive Americans on Medicare.

As...