  Two simple bounds on the $T_c$ of conventional, phonon-mediated
superconductors are derived within the framework of Eliashberg theory in the
strong coupling regime. The first bound is set by the total electron-phonon
coupling available within a material given the hypothetical ability to
arbitrarily dope the material. This bound is studied by deriving a
generalization of the McMillan-Hopfield parameter, $\widetilde{\eta}(E)$, which
measures the strength of electron-phonon coupling including anisotropy effects
and rigid-band doping of the Fermi level to $E$. The second bound is set by the
softening of phonons to instability due to strong electron-phonon coupling with
electrons at the Fermi level. We apply these bounds to some covalent
superconductors including MgB$_2$, where $T_c$ reaches the first bound, and
boron-doped diamond, which is far from its bounds.
