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The PCR test is a very specific test. A positive test means the RNA sequence of SARS-CoV-2 was detected. A negative test, however, would not be as strong of an indicator as false positive rates are quite high early or late in the infection (https://www.biotechniques.com/coronavirus-news/news_false-ne...).

As far as I know a false positive in PCR is not very likely, but of course it is possible.

PCR is very unlikely to give a false positive if you never had the virus. The viral RNA simply isn't present to be detected. A false positive from PCR most likely came from contamination of the test materials.

PCR is quite likely to give a false positive if you previously had the virus but have recovered, including asymptomatic cases. There are still pieces of viral RNA that can be detected by PCR, even if they're inactivated by having destroyed the viral coating and you're no longer contagious.

So PCR is pretty accurate at saying if you ever had the virus, but not so accurate at saying if you currently have it at contagious levels.

Or the antibody test was wrong. Seems most of the tests look super noisy.