EDIT: for those down voting. What about this paper then?
False negative seems extremely plausible. If you're sick, but your viral load is low, it should be possible to collect a sample without a virus particle. If the sample doesn't contain a viral particle, it will necessarily come back negative.
False positive seems unlikely, as you'd have to have something that survives several cloning cycles that ends up looking exactly like the markers you're matching against.
If there are very little viruses, and the swab/sample didn’t contain any you would get a false negative test.
Also, the article doesn't say whether he had fever initially and it went away and then came back, which is important information.