We focus on the problem of computing approximate Nash equilibria and well-supported approximate Nash equilibria in random bimatrix games, where each player's payoffs are bounded and independent random variables, not necessarily identically distributed, but with common expectations. We show that the completely mixed uniform strategy profile, i.e. the combination of mixed strategies (one per player) where each player plays with equal probability each one of her available pure strategies, is an almost Nash equilibrium for random bimatrix games, in the sense that it is, with high probability, an å-well-supported Nash equilibrium where å tends to zero as n tends to infinity.