We analyze the recent claim that experimental measurements of binding energies of dipole-bound anions can be understood in terms of a quantum mechanical anomaly. The discrepancy between the experimental critical dipole moments and that predicted by the anisotropic inverse square potential of a static dipole precludes such an explanation. As has long been known, in the physical problem one must include rotational structure so that the long distance behavior changes from 1/r2 to 1/r4. In a simple model this can be shown to lead to a modification of the critical dipole moment of 20% or so, bringing it into agreement with experiment. This, together with the fact that inclusion of finite size effects does not change the critical dipole moment of the static point dipole, strongly suggests that the quantum mechanical anomaly interpretation of the formation of dipole-bound anions cannot be correct.