Is it normal for someone with costochondritis to feel new chest pains?

Yes: Any problems with the chest (e.g. Pneumonia, bronchitis, costochondritis, heart attack) can result in chest pains. You can't rely on the type of pain to make the diagnosis since there are really no rules how one type of problem should feel. In all cases of chest pain, see a doctor immediately even if turns out to be simply costochondritis. Don't take chances.

Yes: While costochondritis often causes the same type of pain whenever it acts up, it is a type of musculoskeletal pain. Musculoskeletal pains are extremely common in the general population with chest pain, and costochondritis patients are just as likely as anybody else to develop other types of musculoskeletal pain. Other common causes, e.g. Acid reflux, often co-exist as well.