Tesla Inc. Chief Executive Elon Musk, who has repeatedly played down the risk of the coronavirus since early in the pandemic, says he tested both positive and negative for Covid-19 on Thursday and raised questions about the validity of such testing more broadly.

Mr. Musk, on Twitter, said he was experiencing cold-like symptoms and, when taking four of the same tests administered on the same machine, had two results come back positive and two negative.

“Something extremely bogus is going on,” he said.

The symptoms he was experiencing, he said, were “nothing unusual so far.” Later, he tweeted he had experienced “Mild sniffles & cough & slight fever past few days,” though was not feeling any symptoms after taking over-the-counter cold medicine.

If a diagnosis is confirmed, Mr. Musk would become one of several CEOs to reveal they contracted the virus. Morgan Stanley CEO James Gorman, NBCUniversal CEO Jeff Shell and Hewlett-Packard Enterprise Co. CEO Antonio Neri revealed positive diagnoses earlier this year.

Howard Willard, former CEO of Altria Group Inc., took a temporary medical leave after a positive diagnosis in March and announced his retirement in April after a rocky two-year tenure leading the Marlboro maker.

Through the pandemic, management teams have worked to fortify succession plans and review backup operating plans when critical employees fall ill.

In March, Mr. Musk predicted there would likely be close to zero new Covid-19 cases in the U.S. by the end of April. In March, the outspoken CEO said “my guess is that the panic will cause more harm than the virus, if that hasn’t happened already.”

His latest tweets came as positive U.S. Covid-19 tests reached a record. On Thursday, Alameda County, near San Francisco—where Tesla’s lone U.S. car factory is based—warned of rising coronavirus cases and potential new restrictions to combat the disease. County health officials earlier this year ordered Tesla to temporarily close the car plant.

On an earnings call in April, with the U.S. plant shut, Mr. Musk railed against local shelter-in-place restrictions aimed at slowing the spread of the virus.

“Give people back their goddamn freedom,” he said.

When Tesla reopened the Fremont plant in May it had put in place safety protocols that county officials inspected. Workers have reported instances of Covid-19 cases among workers at the facility, though neither Tesla nor local authorities have commented on the scale of infection among the vehicle maker’s staff.

The four Covid-19 results Mr. Musk said he received Thursday came from rapid-response tests. He said he was taking a more thorough PCR, or polymerase chain reaction, test—typically involving a nasal or throat swab—at a separate laboratory and was expecting to get results in about 24 hours.

Rapid-response tests, also known as antigen tests, can deliver results in about 15 minutes but data has shown that they are less precise than PCR tests, meaning they can’t detect the virus as well and may deliver false negatives.

Tesla shares were down about 2% in early-afternoon trade Friday.

The pandemic at one point threatened to derail Mr. Musk’s plan to boost Tesla deliveries by about 36% this year with the closure of the plant. Mr. Musk fought to reopen the factory and Tesla last month said its goal of delivering more than 500,000 vehicles this year could still be attainable. The company is on track to post its first full-year profit in 2020 despite the pandemic, according to analyst estimates.

From the Archives

Aerial footage showed forklifts in action, people walking and cars entering Tesla's car plant in Fremont, Calif., on Monday, as Elon Musk said the electric-vehicle maker would defy a county order and resume production. Photo: David Paul Morris/Bloomberg News (Originally Published May 12, 2020)

Mr. Musk disclosed the test results soon after a rocket being developed by Space Exploration Technologies Corp. that he also runs failed on a test stand in Texas. A different SpaceX rocket awaits a planned Saturday launch at NASA’s Kennedy Space Center in Florida to carry four astronauts to the International Space Station.

—Bowdeya Tweh contributed to this article.

Write to Robert Wall at robert.wall@wsj.com

Corrections & Amplifications
Both Mr. Musk’s Covid-19 tests and Alameda County’s advisory occurred Thursday. An earlier version of this article incorrectly said they occurred Wednesday. (Corrected on Nov. 13)