Atomic and electronic structures of Cu2H and CuH have been investigated by high pressure NMR spectroscopy, X-ray diffraction and ab-initio calculations. Metallic Cu2H was synthesized at a pressure of 40 GPa, and semi-metallic CuH at 90 GPa, found stable up to 160 GPa. Experiments and computations suggest the formation of a metallic 1H-sublattice as well as a high 1H mobility of ~10-7 cm2/s in Cu2H. Comparison of Cu2H and FeH data suggests that deviations from Fermi gas behavior, formation of conductive hydrogen networks, and high 1H mobility could be common features of metal hydrides.