The theoretical analysis of randomized compressive operators often relies on the existence of a concentration of measure inequality for the operator of interest. Though commonly studied for unstructured, dense matrices, matrices with more structure are often of interest because they model constraints on the sensing system or allow more efficient system implementations. In this paper we derive a concentration of measure bound for block diagonal matrices where the nonzero entries along the main diagonal are a single repeated block of i.i.d. Gaussian random variables. Our main result states that the concentration exponent, in the best case, scales as that for a fully dense matrix. We also identify the role that the signal diversity plays in distinguishing the best and worst cases. Finally, we illustrate these phenomena with a series of experiments.