Even as government agencies, companies and academic researchers scramble to validate existing tests and create better ones, there are doubts they can deliver as promised. Most tests now available mistakenly flag at least some people as having antibodies when they do not, which could foster a dangerously false belief that those people have immunity.

And even if the tests do improve, their availability could be hampered by the same manufacturing shortages that have prevented the Covid-19 diagnostic tests from scaling up adequately.

As President Trump presses to reopen the country and several states are considering lifting lockdowns in the next few weeks, widespread screening is considered critical. On Friday, Mr. Trump cheered the F.D.A.’s emergency approval of some antibody tests, saying they would support efforts to get Americans back to work “by showing us who might have developed the wonderful, beautiful immunity.”

Epidemiologists are testing for antibodies in hot spots to better measure the extent of the outbreaks, and government officials intend to use those results to help decide when and how to return residents to daily life. But many scientists and political leaders say the country is nowhere close to deploying enough diagnostic and antibody tests at the speed and volume required.

“The more testing, the more open the economy,” Gov. Andrew M. Cuomo of New York said on Wednesday. He has pushed for the production of antibody tests as a central cog in plans to ease stay-at-home restrictions, saying that New York would eventually screen 100,000 people a day.

Recent testing around the country demonstrates the challenges of using the new products. At the Chicago hospital, for example, the city’s Public Health Department intervened, warning that it should not use antibody tests to determine whether emergency workers were actively infected.

Soon after it helped screen the Rose Law Group, the firm in Arizona, a lab stopped providing rapid tests to other clients, fearing they might not comply with federal guidelines, and switched to more sophisticated lab-based tests.