We fabricate ferromagnetic nanowires with constrictions whose cross section can be reduced gradually from 100 x 30 nm(2) to the atomic scale and eventually to the tunneling regime by means of electromigration. The contacts are mechanically and thermally stable. We measure low-temperature magnetoresistances (MR) <3% for contacts <400 Omega, reproducible MR variations that are nonmonotonic in the regime 400 Omega - 25 kOmega, and a maximum MR of 80% for atomic-scale widths. These results for devices > 400 Omega differ from previous room-temperature studies of electrodeposited devices. For samples in the tunneling regime, we observe large fluctuations in MR, between -10 and 85%.