If this is rare enough, it could fall into the false positive rate of the COVID test they've previously done (i.e. while the test said they had COVID, they didn't)
The tests are designed to have high false positives because they have to bias toward few false negatives and there are no perfect tests. What gets me is that the negative result is the one bearing meaningful information but not the positive result, yet everyone mistakenly talks about positives as if they are far more informative than they actually are.
I'm not sure what test you have in mind, and I'm guessing that such tests with high false positive bias might exist, for example the antibody tests. It's very difficult to get a genuine false positive with a PCR-based test: whatever you're reading out should have the same sequence as the part of the virus you're sequencing. The only reasonable way to get a false positive is that you contaminate the sample, but I don't think that the possibility of contamination is part of the design in any of the commercially available tests.
Certainly the case with the PCR test, because it is so sensitive. It can detect Covid fragments at such a low level that no symptoms would show and no adaptive immune response would arise. That's not enough Covid to generate memory and immunity.
You could be PCR tested positive, while experiencing a cold or flu (which you would now understandably think were Covid symptoms), but would not have immunity for a future, larger infection from Covid.
Though statistically unlikely, it's bound to happen sometimes.
Could be - but the case I remember reading about involved hospitalisation.
Remember that immunisation makes antibodies but only provides something like 50-95% protection (depending on the vaccine) - it's not surprising that actually catching it does the same thing
