Earlier quoted context omitted.
They're highly sensitive to the presence of the target DNA or RNA sequence in the sample well. But they can't easily tell if the target sequence they amplified came from the patient's sample or from other potential sources of contamination in the lab (the test, after all, makes billions of copies of the target DNA sequence), whether from other positive tests or from positive controls.
But that doesn't explain false negatives, does it?
Saw reports that bronchoalveolar lavage (introducing and then suctioning out fluid from a branch of the lung) is better at picking up virus if it's present -- but it's much more time consuming, more invasive, and much less pleasant for the patient than sticking a swab up your nose. Not something you're going to do in a parking lot in a drive through testing center for someone only showing mild symptoms.
There are always compromises.