Questions about the relation between niche overlap measured along single niche dimensions and total niche overlap, and about the relation between overlap and competition, have long troubled ecologists. In this note it is shown that competition may exist even though overall niche overlap in a multidimensional niche space is zero. A distinction is drawn between niche dimensions which describe the various attributes of a single basic resource category, such as food, and niche dimensions which describe functionally distinct ways in which the environment influences the fitness of an organism. Examples of the latter kind of niche dimension include potential limitation by two or more qualitatively distinct resources, and axes describing resource use and escape from predators. The relation between overlap and competition may be complex if niche dimensions reflect qualitatively distinct ways that the environment affects fitness, and, in particular, competition may exist despite zero total overlap in a multidimensional niche space.