We have obtained the mean square displacement for Brownian motion of particles in a fluid under a square-well potential. It is shown that for a deep well, there are short- and long-time regimes where the mean square displacement is proportional to time as well as a long intermediate transition stage. Even for a very mild case where the ratio A of the potential height to the thermal energy is 3 and its width is 5, we need time t of to recover Einstein's relation, which is unpractically too long where D is diffusion coefficient. In the short time regime where an escape process from the well dominates the considerably slow dynamics, the mean square displacement is approximately given by with the exponential factor appearing in theory of chemical reactions.