But that real information is now telling the state’s leadership a story it most likely did not want to hear: As in much of the South and West, Tennessee is awash in confirmed cases, and testing has proved no match for the coronavirus once it overwhelms local governments’ abilities to trace an infected person’s contacts and forces those who were exposed to self-quarantine.

Tennessee is far from the only state to discover that despite Mr. Trump’s hype — he boasted on Monday on Twitter: “our great testing program continues to lead the World, by FAR!” — coronavirus testing is not a miracle path to a safe reopening. As the nation faces a new shortage of tests, Tennessee’s experiences offer a cautionary tale about the limits of testing.

In Georgia, Gov. Brian Kemp, a Republican, expanded testing in early May and urged all residents to make appointments; in Oklahoma, Gov. Kevin Stitt, a Republican, did much the same thing. In Ohio, Gov. Mike DeWine announced on June 11 that any Ohioan could get a test; he and his wife, Fran DeWine, along with the state’s lieutenant governor, Jon Husted, took tests at a news conference as a way of encouraging others to do so. Each state has since had a sharp increase in confirmed infections.

“This was just as predictable as buying snow tires in June for your car in Minnesota,” said Dr. Michael T. Osterholm, the director of the Center for Infectious Disease Research and Policy at the University of Minnesota. “You know December is coming.”