The effect of salt on the dynamics of water molecules follows the Hofmeister series. For some "structure-making" salts, the self-diffusion coefficient of the water molecules, D, decreases with increasing salt concentration. For other "structure-breaking" salts, D increases with increasing salt concentration. In this work, the concentration and temperature dependence of the self-diffusion of water in electrolyte solutions is studied using molecular dynamics simulations and pulsed-field-gradient NMR experiments; temperature-dependent viscosities are also independently measured. Simulations of rigid, nonpolarizable models at room temperature show that none of the many models tested can reproduce the experimentally observed trend for the concentration dependence of D; that is, the models predict that D decreases with increasing salt concentration for both structure-breaking and structure-making salts. Predictions of polarizable models are not in agreement with experiment either. These results suggest that many popular water models do not accurately describe the dynamic nature of the hydrogen bond network of water at room temperature. The simulations are in qualitative agreement, however, with experimental results for the temperature dependence of water dynamics; simulations and experiment show an Arrhenius dependence of D with temperature, T, with added salt, that is, In D similar to 1/T, over a range of temperatures above the freezing point of water.