This work describes an approximate and temporary form of ergodicity, called pseudoergodicity. This is a classical phenomenon that is of particular interest since it has a straightforward quantum analog and can be expected to influence the dynamics of a quantum system more directly than rigorous ergodicity. A procedure is described for locating phase space regions where classical motion is pseudoergodic (pe) and this technique is applied to the Henon-Heiles system excited to near its escape energy. Certain pe zones containing chaotic trajectories appear to coincide with the vague tori of Reinhardt and co-workers while other such zones have less familiar forms. When the Henon-Heiles system is observed for long times, a single pe region becomes larger than others, thus marking incipient ergodicity on much of the energy shell. The classical pe regions are compared to quantum pe zones, calculated by a slight modification of a technique presented in an earlier paper. It is found that certain classical pe regions have close quantum analogs while others do not, for reasons probably related to the short time scale for quantum-classical correspondence. Among the classical regions that do not have quantum analogs is the aforementioned dominant pe zone. The implication is that the present quantum system does not display behavior similar to the truly large-scale ergodicity that occurs in the classical system.