Yes, each testing methodology undergoes numerous tests and has a known margin of error. False positives are rare and are usually from cross contamination, and that usually involves the robotic or multi open tube tests. This is what happened to the NFL.
Using a single test system like the Genxpert vastly reduces the false positives because cross contamination is much more difficult to happen.
If the testing system has a higher than acceptable false positive rate, the FDA will issue a warning, like it did in July when the BD SARS-CoV-2 Reagents caused up to a 3% false positive rate.
So yes, false positives happen, but even with incidents like the 77 NFL players, it is still a rare occurrence. False negatives are still higher, possibly due to incorrect nasal swabbing. The saliva tests may be promising, but the covid-19 seems more concentrated in the nasal passages.
With anything medical, always ask questions and read to make an informed decision.