This article focuses on the development of distributed robust model predictive control (MPC) methods for multiple connected and automated vehicles (CAVs) to ensure their safe operation in the presence of uncertainty. The proposed layered control framework includes reference trajectory generation, distributionally robust obstacle occupancy set computation, distributed state constraint set evaluation, data-driven linear model representation, and robust tube-based MPC design. To enable distributed operation among the CAVs, we present a method, which exploits sampling-based reference trajectory generation and distributed constraint set evaluation methods, that decouples the coupled collision avoidance constraint among the CAVs. This is followed by data-driven linear model representation of the nonlinear system to evaluate the convex equivalent of the nonlinear control problem. Finally, to ensure safe operation in the presence of uncertainty, this article employs a robust tube-based MPC method. For a multiple CAV lane change problem, simulation results show the efficacy of the proposed controller in terms of computational efficiency and the ability to generate safe and smooth CAV trajectories in a distributed fashion.