It is a mainstay of community ecology that local exclusion of species will result if competitive pressures become too large. The pattern of exclusion may be complicated, but the qualitative orthodoxy has changed little since the pioneering work of Lotka, Volterra, and Gause--no two species can occupy the same niche. Stated in a more precise form, the higher the intensity of interspecific competition in an assemblage of species, the fewer the number of species that can coexist in perpetuity. We suggest that this orthodoxy results from "linear" thinking, and that if the classical equations are formulated more realistically with attendant nonlinearities, the orthodoxy breaks down and higher levels of competition may actually increase the likelihood that species will avoid competitive exclusion. Furthermore, this increased probability of coexistence at higher levels of competition is accompanied by characteristic dynamic patterns: (i) at lower levels of competition, after all extinction events have occurred, remaining species follow irregular chaotic patterns; (ii) at higher levels of competition, when most species coexist, all species are entrained in a single large limit cycle; (iii) the transient behavior appears to correspond to a special case of chaos, uniform phase chaotic amplitude.