The accuracies of two theoretical expressions for thermal boundary resistance are assessed by comparing their predictions to independent predictions from molecular dynamics (MD) simulations. In one expression $({R}_{E})$, the phonon distributions are assumed to follow the equilibrium, Bose-Einstein distribution, while in the other expression $({R}_{NE})$, the phonons are assumed to have nonequilibrium, but bulk-like distributions. The phonon properties are obtained using lattice dynamics-based methods, which assume that the phonon interface scattering is specular and elastic. We consider (i) a symmetrically strained Si/Ge interface, and (ii) a series of interfaces between Si and ``heavy-Si,'' which differs from Si only in mass. All of the interfaces are perfect, justifying the assumption of specular scattering. The MD-predicted Si/Ge thermal boundary resistance is temperature independent and equal to $3.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}\text{ }{\text{m}}^{2}\text{-K}/\text{W}$ below a temperature of $\ensuremath{\sim}500\text{ }\text{K}$, indicating that the phonon scattering is elastic, as required for the validity of the theoretical calculations. At higher-temperatures, the MD-predicted Si/Ge thermal boundary resistance decreases with increasing temperature, a trend we attribute to inelastic scattering. For the Si/Ge interface and the Si/heavy-Si interfaces with mass ratios greater than two, ${R}_{E}$ is in good agreement with the corresponding MD-predicted values at temperatures where the interface scattering is elastic. When applied to a system containing no interface, ${R}_{E}$ is erroneously nonzero due to the assumption of equilibrium phonon distributions on either side of the interface. While ${R}_{NE}$ is zero for a system containing no interface, it is 40%--60% less than the corresponding MD-predicted values for the Si/Ge interface and the Si/heavy-Si interfaces at temperatures where the interface scattering is elastic. This inaccuracy is attributed to the assumption of bulk-like phonon distributions on either side of the interface.