Roughly 50% of all first-ranked cluster galaxies are seen with two or more nuclei within 20 kpc of the luminosity center of the the system (H0 = 50 km s-1Mpc-1). Spectroscopic measurements and CCD imaging are presented for the nuclear components of four multiple-nucleus systems: those found in Abell clusters 400, 671, 1185, and 2052. The spectroscopic and imaging data for the systems in the two clusters A400 and A671 suggest that their multiplicity is no more than a projection effect. The shapes of the contours around the nuclear components in A1185 and A2052 are consistent with physical association among those nuclei, a conjecture supported by the spectroscopic data. This physical association has probably resulted from the action of dynamical friction and the subsequent orbital decay of cluster galaxies whose trajectories brought them close to the first-ranked galaxy.