“I’m terrified,” said Mr. Suarez, who has been a paramedic in New York City for 26 years and had assisted in rescue efforts during the Sept. 11, 2001, terror attacks and later served in the Iraq war. “I honestly don’t know if I’m going to survive. I’m terrified of what I’ve already possibly brought home.”

Even as hospitals across New York become inundated with coronavirus cases, some patients are being left behind in their homes because the health care system cannot handle them all, according to dozens of interviews with paramedics, New York Fire Department officials and union representatives, as well as city data.

In a matter of days, the city’s 911 system has been overwhelmed by calls for medical distress apparently related to the virus. Typically, the system sees about 4,000 Emergency Medical Services calls a day.

On Thursday, dispatchers took more than 7,000 calls — a volume not seen since the Sept. 11 attacks. The record for amount of calls in a day was broken three times in the last week.