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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 10 10 /D 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 4e −A Dt with the exponential factor appearing in theory of chemical reactions.