We measure the low-temperature resistance of permalloy break junctions as a function of contact size and the magnetic field angle in applied fields large enough to saturate the magnetization. For both nanometer-scale metallic contacts and tunneling devices we observe large changes in resistance with the angle, as large as 25% in the tunneling regime. The pattern of magnetoresistance is sensitive to changes in bias on a scale of a few mV. We interpret the effect as a consequence of conductance fluctuations due to quantum interference.