All current phylogenetic methods assume that DNA substitutions are independent among sites. However, ample empirical evidence suggests that the process of substitution is not independent but is, in fact, temporally and spatially correlated. The robustness of several commonly used phylogenetic methods to the assumption of independent substitution is examined. A compound Poisson process is used to model DNA substitution. This model assumes that substitution events are Poisson-distributed in time and that the number of substitutions associated with each event is geometrically distributed. The asymptotic properties of phylogenetic methods do not appear to change under a compound Poisson process of DNA substitution. Moreover, the rank order of the performance of different methods does not change. However, all phylogenetic methods become less efficient when substitution follows a compound Poisson process.