An electronic device may have magnetically mounted antenna structures. The electronic device may have a dielectric member against which one or more antennas are mounted. The dielectric member may be a cover glass layer that covers a display in the electronic device, a dielectric antenna window, or other dielectric structure. Each antenna may have an antenna support structure. Conductive antenna structures for the antenna may be mounted to the antenna support structure. The antennas may be cavity-backed planar inverted-F antennas. Portions of each antenna support structure may be configured to receive magnets. The magnets may be attracted towards ferromagnetic structures mounted on the dielectric member. As the magnets are attracted towards the ferromagnetic structure, the antennas may be held in place against the dielectric member.