But these patients are still too sick to go home, too sick even for most nursing homes.

“It’s truly a hidden segment, even to most people in health care,” said Dr. Anil Makam, a hospitalist and researcher at the University of California, San Francisco.

Take the man in his 60s slowly making his way down the hallway, supported by a large wheeled apparatus that had also lifted him out of bed. Two physical therapists and two aides accompanied him.

He had been in the nonprofit Hospital for Special Care for about a month, on top of 30 days in a standard hospital, trying to recover from a brain hemorrhage and spinal cord injuries after a fall down the stairs in his home.

He had finally come off the ventilator the week before, although he might need it again. “Setbacks are very common,” said Dr. John Votto, a pulmonologist and former chief executive of the hospital, watching the patient’s progress.

In one cubicle lay a woman with severe emphysema who had endured six hospitalizations in five months. “She’s off the vent, but she’s very tenuous, in and out, in and out,” Dr. Votto said. She hadn’t been home since January.