The classical theory of the electrostatic persistence length of polyelectrolyte chains due to Odijk, Skolnick and Fixman (OSF) is revisited. In contrast to the original theory, our approach takes into account the effect of fluctuations in the chain configuration. As a result, we find that the OSF theory, although correct for intrinsically rigid polyelectrolytes, breaks down for flexible chains. A quantitative criterion is established to distinguish between flexible and rigid chains. For flexible chains, we use a variational theory to compute the electrostatic persistence length, which is found to be proportional to the Debye-Hückel screening length, in agreement with many experimental and simulation results.