A spokeswoman for the Association of Public Health Laboratories, Michelle Forman, said there were about 72 public health laboratories that are presently testing for the new coronavirus. “We are not aware of any widespread testing shortages,” she said.

For weeks, New York and other cities around the nation had been hampered in their ability to rapidly test people for the coronavirus. All testing was initially done by the C.D.C. in its headquarters in Atlanta, with narrow criteria for who qualified.

People who returned from abroad with possible symptoms have often been rebuffed when they sought testing. Doctors and hospitals have also faced an uphill battle with the government when they try to get patients tested for the coronavirus.

There were also problems with faulty test kits being distributed by the C.D.C. And this week, one nurse in California who is sick after treating a coronavirus patient released a statement describing the difficulties she has had trying to get tested.

On Thursday, Vice President Mike Pence, who previously vowed that “any American could be tested,” conceded that “we don’t have enough tests today to meet what we anticipate will be the demand going forward.”

New York has now begun its own tests: Last weekend, the state’s public health laboratory in Albany, the Wadsworth Center, received federal approval to begin testing.

On Monday, Gov. Andrew M. Cuomo said he hoped that within the week the state and city would be conducting 1,000 tests a day.