Given a group G, the model G(G,p) denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. In this article we show that for any epsilon>0 and any family of groups G(k) of order n(k) for which nk, a graph kG(Gk,p) with high probability has diameter at most 2 if p(2+epsilon)lognknk and with high probability has diameter greater than 2 if p(14+epsilon)lognknk. We also provide examples of families of graphs which show that both of these results are best possible. Of particular interest is that for some families of groups, the corresponding random Cayley graphs achieve Diameter 2 significantly faster than the Erds-Renyi random graphs.