Tunneling involves an allowed quantum event which fails to take place classically. Dynamical tunneling is the subset of such events which do not involve a classically insurmountable potential barrier. In this paper, we present unambiguous evidence for dynamical tunneling in bound state quantum systems.The tunneling occurs between two distinct regions of classically trapped quasiperiodic motion. Close analogies are shown to exist between this situation and ordinary barrier penetration in a double minimum potential. In the cases we study, tunneling occurs between equivalent or nearly equivalent local mode motions, which have arisen out of a resonance between the symmetric and nonsymmetric stretch.