Federal health officials announced on Wednesday that anyone who wants a coronavirus test may get one if a doctor agrees. But the nation’s testing capacity is still so limited that experts feared clinics and hospitals could be overwhelmed by an avalanche of requests.

Under the new criteria, patients who have fevers, coughs or difficulty breathing qualify for diagnostic testing, depending on their doctor’s judgment. But with flu season in full swing, tens of millions of Americans already have respiratory symptoms, and doctors have no quick way to discern who should be tested.

The Trump administration has repeatedly promised to expand the nation’s testing capabilities by Friday, even as state laboratories estimated that it would be weeks before millions of American could be tested.

“We are already overwhelmed with calls day and night,” said Dr. Julie Vaishampayan, public health officer of Stanislaus County in Modesto, Calif. Public health laboratories in California are already working seven days a week, she said.