We present the results of our multiwavelength study of two nearby Abell clusters that contain extended tailed radio sources. From our analysis of archival PSPC x-ray data, VLA radio observations, and new velocity measurements, we find strong evidence that these clusters are not relaxed. Significant x-ray substructure is revealed by examining deviations from circular models of the overall x-ray surface brightness maps. We report 130 new redshifts from the fields of A578 and A1569 and find that 41 galaxies belong to A578 and 54 belong to A1569. Moreover, we detect the presence of substructure in the velocity and spatial distribution of galaxies. We identify two interacting subclusters in A578 and two gravitationally unbound subclusters in A1569. Furthermore, we find that the radio sources in A578 are at rest with respect to their subclusters and that the wide-angled tailed (WAT) source in A1569 has a small peculiar motion with respect to its subcluster (220 km/s). The standard model of ram pressure induced jet curvature requires relative velocities of the radio sources with respect to the ICM of greater than or similar to 1000 km/s. Therefore, the apparent peculiar velocities of the radio galaxies cannot explain the bending of the jets/tails for the narrow-angled tailed (NAT) radio source in A578 and for the WAT in A1569. We suggest that a cluster-subcluster merger model provides a reasonable explanation for the x-ray and optical substructure detected in these systems. Numerical simulations of cluster-subcluster mergers predict a bulk flow of ICM gas that has sufficient dynamic pressure for bending and shaping the extended radio sources. Thus, a cluster-subcluster merger can provide the environment responsible for the observed cluster x-ray and optical morphologies and for the bending seen in the radio sources. (C) 1997 American Astronomical Society.