Recent experiments have found that the anisotropic magnetoresistance (AMR) of nanometer-scale ferromagnetic contacts at low temperature can be larger than that of bulk samples and can exhibit more complicated variations as a function of sample bias and the angle of an applied magnetic field than in the bulk case. Here, we test a proposal that quantum interference of electrons may explain these results, by measuring the temperature dependence of the AMR signals in nanometer-scale contacts made from Permalloy (Ni80Fe20), Ni, and Co. We find a strong temperature dependence, in quantitative agreement with expectations for a quantum-interference effect. In the course of making these measurements, we also observed that two-level resistance fluctuations as a function of time, associated with reconfigurations of the atomic structure, are present in all of our samples as the temperature is increased above a few tens of Kelvin, and they can be found in some samples even at 4.2 K. The relative energy of the different atomic configurations can be extremely sensitive to the angle of the sample magnetization, so that the conductance in some samples can be made to change abruptly and reproducibly as the angle of an applied magnetic field is rotated.