They typically do one swab of the nasopharynx (upper back of the throat, up above the soft palate). You might have the virus elsewhere but not there.

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.

It depends on the time of the illness. Roughly my understanding is: 1st week - high virus load in the nasopharyngeal area. 2nd - almost no virus load in the nasopharyngeal area, high virus load in the lungs