Recent publications have drawn attention to the idea of utilizing prior information about the correlation structure to improve statistical power in cluster randomized experiments. Because power in cluster randomized designs is a function of many different parameters, it has been difficult for applied researchers to discern a simple rule explaining when prior correlation information will substantially improve power. This article provides bounds on the maximum possible improvement in power as a function of a single parameter, the number of clusters at the highest level of a multilevel experiment. The maximum improvement in power is less than 0.05 unless the number of clusters at the highest level is less than 20. Thus, the utility of using prior correlation information is limited to experiments with very small cluster-level sample sizes. Situations where small cluster-level sample sizes could still result in experiments with good statistical power are discussed, as is the relative utility of prior information about intracluster correlations as compared with covariate information that can explain cluster level variability in the outcome.