Musty Smell in Apt

Every time it rains, my apartment gets a slight musty smell that requires opening most of the windows to get it out. We don’t have any leaks (that we know of, at least), and when we opened the walls to renovate in December there wasn’t any evidence of mold or leaky pipes in the walls. The building is old (almost 100 yrs) but there doesn’t seem to be any water damage anywhere. Any ideas? The smell is the worst near the bathroom sink, in the coat and linen closet, and in a small section of the bedroom.