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We describe computational experiments exploring the performance improvements from overlapping computation and communication on hybrid parallel computers. Our test case is explicit time integration of linear advection with constant uniform velocity in a three-dimensional periodic domain. The test systems include a Cray XT5, a Cray XE6, and two multicore Infiniband clusters with different generations of NVIDIA graphics processing units (GPUs). We describe results for Fortran implementations using various combinations of MPI, OpenMP, and CUDA, with and without overlap of computation and communication. We find that overlapping CPU computation, GPU computation, parallel communication, and CPU-GPU communication can provide performance improvements of more than a factor of two.