Using molecular dynamics computer simulation, we have calculated the velocity autocorrelation function and diffusion constant for a spherical solute in a dense fluid of spherical solvent particles. The size and mass of the solute particle are related in such a way that we can naturally approach the Brownian limit (when the solute becomes much larger and more massive than the solvent particles). We find that as long as the solute radius is interpreted as an effective hydrodynamic radius, the Stokes-Einstein law with slip boundary conditions is satisfied as the Brownian limit is approached (specifically, when the solute is roughly 100 times more massive than the solvent particles). In contrast, the Stokes-Einstein law is not satisfied for a tagged particle of the neat solvent. We also find that in the Brownian limit the amplitude of the long-time tail of the solute's velocity autocorrelation function is in good agreement with theoretical hydrodynamic predictions. When the solvent density is substantially lower than the triple density, the Stokes-Einstein law is no longer satisfied, and the amplitude of the long-time tail is not in good agreement with theoretical predictions, signaling the breakdown of hydrodynamics. (C) 2003 American Institute of Physics.