We have identified anomalous behavior of the escape rate out of the zero-voltage state in Josephson junctions with a high critical current density J(c). For this study we have employed YBa2Cu3O7-x grain boundary junctions, which span a wide range of J(c) and have appropriate electrodynamical parameters. Such high Jc junctions, when hysteretic, do not switch from the superconducting to the normal state following the expected stochastic Josephson distribution, despite having standard Josephson properties such as a Fraunhofer magnetic field pattern. The switching current distributions (SCDs) are consistent with nonequilibrium dynamics taking place on a local rather than a global scale. This means that macroscopic quantum phenomena seem to be p...