One woman in Harlem who wanted to be tested was told by health care workers not to worry about her coronavirus-like symptoms. In Brooklyn, a woman had to wait to get tested until her mother tested positive for the virus. One doctor at a hospital network has turned away patients who probably had the coronavirus because they did not meet the current testing criteria.

President Trump has proclaimed that anyone who wants to get a coronavirus test can. Gov. Andrew M. Cuomo has declared that the state should be testing 1,000 people a day.

In New York, at least, the reality seems far more complicated.

Numerous interviews with doctors, hospital administrators and health officials this week revealed a confused and often troubled testing system in New York that has left many people who believe that they have been exposed to the coronavirus deeply frustrated.

The accounts suggest that the state and local governments have at times struggled to roll out one of the most effective means for responding to the epidemic: widespread testing.