We apply the ab initio no-core nuclear shell model to solve the six-nucleon systems, (6)Li and (6)He, interacting by realistic nucleon-nucleon ( NN) potentials. In particular, we present the first results for A = 6 with the nonlocal CD-Bonn NN potential. The resulting (6)Li binding energy -29.3(6) MeV and the excitation spectra improve the agreement between the theory and experiment compared to results with local NN potentials, but the need for the inclusion of a real three-body interaction and/or further improvement of the NN forces remains. We predict properties of the (6)He dipole modes, a subject of current controversy.