It's possible to have had COVID and not have antibodies. What we don't know exactly is the percentage of people who may be in this situation, but it can happen because Antibodies are just one of the pathways that the immune system uses to fight infections, but there are other ways to do so. First, we have the innate response, which is a very rapid, nonspecific response. It's the body's first reaction to an infection from any type of pathogen, not just viruses. Sometimes, if the innate response is very strong, the body may prevent us from becoming infected, or if we do, the infection is quickly contained. In this case, the person to whom this happens would have virtually no symptoms because this response is very rapid, and they wouldn't necessarily develop antibodies despite having been infected.
