Abstract.   In this paper we develop a mathematical model in which any number of competing species can coexist on four resources which regenerate according to an algebraic relationship. We show that previous attempts to prove that n species cannot coexist on fewer than n resources (the “competitive exclusion principle”) all make use of the very restrictive assumption that the specific growth rates of all competing species are linear functions of resource densities. When this restriction is relaxed, it becomes possible to find situations in which n species can coexist on fewer than n resources. On the basis of this and other observations we conclude that the competitive exclusion principle should be considered to apply only to coexistence at fixed densities.