In a nonrelativistic theory, the “contact” interaction between two point particles has a delta-function singularity. It is shown for the attractive case that this leads to an infinite binding energy for the ground state, and hence cannot be physically meaningful. The hyperfine shift in the energy is calculated, using the variational principle, for a smeared-out interaction both for hydrogen and positronium. It is found that the binding energies become unphysically large if the sizes of the particles are reduced beyond a certain nonzero limit which represents the minimum size of the particle.