Among 39 positive antigen test results from both BD and Quidel, 23 were found by P.C.R. to be negative — an error rate of nearly 60 percent.

The results, which were collected from a dozen facilities where thousands of tests had been performed, prompted the state to pivot away from antigen tests to viral RNA tests such as P.C.R., according to the directive. Some nursing homes will also switch to another rapid test called Abbott ID NOW, which has been frequently used by the White House to screen visitors and staff in close contact with President Trump; this test, too, has produced unreliable results.

The number of tests evaluated in Nevada was small, and does not necessarily represent the performance of the antigen tests in other contexts. In communities where the coronavirus is more widespread, for instance, the number of true positives might exceed the number of false positives.

But Susan Butler-Wu, a clinical microbiologist at the University of Southern California, noted that Nevada’s data could be emblematic of a larger issue: the off-label use of tests that are designed and validated for certain populations, but that are then administered to others.

Both BD’s and Quidel’s tests received F.D.A. clearance for use “within the first five days of the onset of symptoms.” The instructions that come with BD’s test have noted that “the performance of this test has not been evaluated for use in patients without signs and symptoms of respiratory infection and performance may differ in asymptomatic individuals.”

In late August, the Department of Health and Human Services began requiring nursing homes to routinely test their residents and staff, including people without symptoms. Days later, the agency extended coverage under the Public Readiness and Emergency Preparedness Act to include screening for infections in asymptomatic people in nursing homes and other congregate facilities.

The measures were implemented to quash outbreaks among nursing home residents, who tend to be at high risk for infection and serious symptoms if exposed to the virus. But experts have repeatedly warned that in the absence of data demonstrating that the tests can accurately evaluate people with no symptoms, results should not be taken as definitive.

Nevada’s recent testing woes reaffirm “why you can’t take something that’s approved for symptomatic use and apply it to an asymptomatic population,” Dr. Butler-Wu said.

Lisa Sanders, director of media relations at LeadingAge, an association of nonprofit providers of aging services, said several nursing homes in other states had been experiencing issues with BD and Quidel’s tests and reporting them to her organization and the American Health Care Association in recent weeks. Overall, false positives were rare. But more than two dozen facilities reported more than six discordant results — enough to “warrant an investigation,” according to a September report by the A.H.C.A. and LeadingAge.

“There was an immediate effort to find out what’s going on,” Ms. Sanders said. Concern among her colleagues has been high, she added: “If the tests aren’t working, what are you going to do?”

Peter Iwen, director of the Nebraska Public Health Laboratory, said facilities in his state had also collected data suggesting that BD’s test, called the Veritor, yielded a concerning number of false positives. Once further results on these and other rapid tests become available, he added, “we may be shocked at how inaccurate the Covid-19 antigen tests are.”

Quidel’s rapid antigen tests have come under fire before. In August, a testing location in Manchester, Vt., reported a rash of false positives produced by the company’s product, called the Sofia.

In response, Douglas Bryant, president and C.E.O. of Quidel Corporation, told reporters it was “highly likely” the errors had arisen not with the company’s antigen tests but with the P.C.R. tests used to confirm them, which were “at risk of providing inaccurate results.”

When contacted about the Nevada directive, Jeannine Sharp Mason, Quidel’s director of marketing communications, said the company was aware of the situation and remained “confident in the accuracy of our rapid antigen tests.” She added that Quidel was “engaging with the state” to “determine the root cause of the alleged discordance.”

Asked to clarify whether off-label use could have prompted false positives, Ms. Mason declined to comment further, adding only that “nothing at this time tells us there is an issue with the product.”

Kristen Cardillo, BD’s vice president of global communication, said the company was aware of the situation in Nevada and was “conducting thorough investigations.” She added that “based on the information in the directive and the total tests performed, we believe the rate of reported false positives is well within what we would expect for the BD Veritor System.”

Representatives for the Department of Health and Human Services did not respond to requests for comment.