This paper weakens the size and moment conditions needed for typical block bootstrap methods (i.e. the moving blocks, circular blocks, and stationary bootstraps) to be valid for the sample mean of Near-Epoch-Dependent functions of mixing processes; they are consistent under the weakest conditions that ensure the original process obeys a Central Limit Theorem (those of de Jong, 1997, Econometric Theory). In doing so, this paper extends de Jong's method of proof, a blocking argument, to hold with random and unequal block lengths. This paper also proves that bootstrapped partial sums satisfy a Functional CLT under the same conditions.