In fact, several months after we were vaccinated this did happen to my gf and I after we were exposed to Covid on a trip to see one of her non-vaxing relatives. The week after we returned home we both started testing positive. We isolated for a week, but developed no symptoms. Coronaviruses reside in your nose while you test positive, so you could possibly pass it on during the positive period.
Testing, checking contacts and isolation during a pandemic are all sciency things that my spoiled generation resists because we have to put a. certain amount of thought and effort into fighting a deadly virus, and we were not about to go to the trouble.
If my generation had been in charge during WW II, we would all be speaking German today.