Yes, that's the hypothesis of the "There's no Covid-19" crowd that will disappear whenever hospital corridors are filled with dead bodies.
On the other hand PCR test are not inaccurate at all, at least regarding false positives. There are detailed articles and even Youtube videos on the process and if you take a look, you will understand why it's quite accurate.
In essence, they create a reaction that will mass produce specific parts of the virus gene until it's detectable with a sensor. They run the reaction multiple times and then the sensor looks for luminance if I recall correctly.
Because you can't accidentally produce Covid-19 genes, the tests are very accurate. However, you might not get enough virus material when taking the sample from the patient(which can fail your reaction or you might need more cycles than usual to create a detectable amount) or it could be the case that there's a mutation in the gene that you are attempting to multiply, therefore you can get a negative test for an infected person.