Theories such as the mode coupling theory (MCT) have seen recent success in predicting the kinetic arrest boundaries and resultant flow properties of colloidal suspensions. A key assumption of such theories is that interparticle forces and equilibrium structure control slow dynamics and gelation, not long-time many-body hydrodynamics. Here we report measurements of short-time collective diffusivities of colloid-polymer suspensions aimed at elucidating the relative contributions of hydrodynamics and thermodynamics as a phase transition or gelation boundary is approached. The experimental system is a hard sphere octadecyl silica suspension to which nonadsorbing polystyrene is added. Two different polymer molecular weights are chosen such that they give rise to a liquid-liquid or a gel transition as the colloid volume fraction or polymer concentration is increased. The short-time diffusivities are measured for each polymer molecular weight as a function of polymer concentration and colloid volume fraction. At a fixed polymer molecular weight and concentration, the colloid volume fraction is varied from dilute to concentrated and near the phase separation boundary. It is found for all measured colloid volume fractions that the diffusivities decrease linearly with increasing strength of the polymer-mediated depletion attraction at a fixed polymer molecular weight. Comparisons are made with theoretical predictions in the dilute limit. When the effects of thermodynamics are normalized out by multiplying the measured diffusivities with the suspension structure factor, it is found that the hydrodynamic effects are essentially those of hard spheres independent of the range and strength of depletion attraction.