As I posted earlier in this thread, I have personally had a *double* positive antigen test (on the same day, same occasion), followed up by a negative PCR test but this is a very rare occurrence.
False negatives are way more common than false positive antigen tests. The hospital I am working at is currently conducting hundreds of antigen tests per week and false positives are extremely rare while there are always people who slip through the antigen test and only show up positive when PCR comes into play.
From personal experience: I have gone to a test center almost daily in December in spite of being fully vaccinated (and boostered since 12/30) because I needed the tests for nursing home visits and for work.
The quality of the actual testing varies A LOT in my experience but it is generally pretty poor. It depends a lot on the person who is performing the test.
The way the test (instructions for the very common Roche test) is supposed to be conducted is that the test swab should be inserted into both nostrils, turned at least four times and remain inside each nostril for about 15 seconds.
With the Roche test it is not even required that the swab is inserted deeply into the nostrils like with other tests. Only 2cm is sufficient.
However, the test center that I still go to several times per week perform a classic invasive deep nasal swab. They only swab one nostril and most of the time the swab is inserted for only a maximum of about five seconds. Sometimes the test is performed extremely poorly with a quick in and out movement and barely any turning of the test swab.
I would say that at least about 70% of the tests they performed on me were near useless and, if I would have been infected with CoVid, the tests probably only would have turned up a positive result if I would have been like super-infected with actually visible virus or something
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TL;DR I would definitely say that antigen tests are quite unreliable and that only a fraction of actual infections are detected due to poor test execution. False positives, however, are extremely rare, but it can happen… like it did to me twice in a row for whatever weird reason.