Theoretical implications of the suggestion that the observed nucleon electromagnetic form factors indicate the existence of a nucleon "core" are discussed. On the basis of physical arguments concerning the nature of such a core, it is shown that, for the neutron, both the charge form factor, ${{F}_{\mathrm{ch}}}^{n}({q}^{2})$, and the magnetic form factor, ${{F}_{\mathrm{mag}}}^{n}({q}^{2})$, must vanish as ${q}^{2}$, the invariant momentum transfer, increases without limit. On the other hand, for the proton ${{F}_{\mathrm{ch}}}^{p}({q}^{2})\ensuremath{\rightarrow}{{Z}_{2}}^{(s)}$ and ${{F}_{\mathrm{mag}}}^{p}({q}^{2})\ensuremath{\rightarrow}\frac{{{Z}_{2}}^{(s)}}{2M}$, where ${{Z}_{2}}^{(s)}$ is the wave function renormalization constant for strong interactions, which is a measure of the probability of the "core state." In terms of the Dirac form factor, ${F}_{1}({q}^{2})$, and the Pauli form factor, ${F}_{2}({q}^{2})$, these results read ${{F}_{1}}^{n}({q}^{2})\ensuremath{\rightarrow}0$, ${{F}_{1}}^{p}({q}^{2})\ensuremath{\rightarrow}{{Z}_{2}}^{(s)}$, and ${q}^{2}{F}_{2}({q}^{2})\ensuremath{\rightarrow}0$ for both neutron and proton. The results for ${F}_{1}({q}^{2})$ are the same as those obtained by Hiida, Nakanishi, Nogami, and Uehara. The other result implies the existence of a relationship which may be used to eliminate one parameter in the analysis of ${F}_{2}$. The generality of the interpretation of ${F}_{\mathrm{ch}}$ and ${F}_{\mathrm{mag}}$ as Fourier transforms of distributions of charge and magnetization, respectively, is demonstrated in the Appendix.