This article investigates the relationship between the similarity of resource capture abilities and the amount of competition between two consumer species that exploit common resources. Most of the analysis is based on a consumer-resource model introduced by Robert MacArthur. Contrary to many statements in the literature and in textbooks, measures of competition may decrease as similarity increases and may be greatest when similarity of the two species' sets of resource capture rates is very low. High competition with low similarity may occur whether competition is measured by a competition coefficient near equilibrium or is measured by the proportional increase in a species' population density when its competitor is removed. However, these two measures may differ considerably and may change in opposite directions with a given change in similarity. The general conditions required for such counterintuitive relationships between similarity and competition are that the consumer species have relatively low resource requirements for successful reproduction and that the resources be self-reproducing. These same conditions also frequently lead to exclusion of one or more resources via apparent competition, and this is always true of MacArthur's model. A variety of other models of competition are analyzed, and circumstances most likely to produce large competitive effects with little overlap are identified.