Any test can have false positives. An article at The Conversation suggests, based on Australian data, that the RT-PCR test has a false positive rate of about 4 percent. Estimates of false negatives range from 1.8 all the way to 58 percent, but a lot of those numbers come from low-quality studies, so the best we can say is that we don't really know.
The thing is, COVID is not highly prevalent; to this day, most people have never contracted it. Of those countries where good records are available, the highest rate of cumulative cases is in the neighbourhood of 20 percent. Of course, the active cases at any given time are far fewer.
In Canada, about 1.8 million cases have been recorded, with about 33,000 being active at present. That second number is 0.1 percent of the population. If you tested everybody and had a 4 percent false-positive rate, the false positives alone would be nearly as many as the cumulative total number of cases, and over 40 times as many as the actual number of active cases! (In fact, the rate of positive tests is running at about 3.7 percent, which is almost exactly what you would find if everyone was COVID-negative. This suggests that a single COVID test is not a very useful diagnostic instrument.)
So no, they're not impossible, and they're not even rare. But it really won't do for the authorities to admit as much, because they are basing their policies on the assumption that all positive tests represent real COVID cases. A good many of the official proclamations, on this as on other subjects, are coloured by officials' need to deny their mistakes or cover their arses.
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