Networks commonly exhibit a community structure, whereby groups of vertices are more densely connected to each other than to other vertices. Often these communities overlap, such that each vertex may occur in more than one community. However, two distinct types of overlapping are possible: crisp (where each vertex belongs fully to each community of which it is a member) and fuzzy (where each vertex belongs to each community to a different extent). We investigate the effects of the fuzziness of community overlap. We find that it has a strong effect on the performance of community detection methods: some algorithms perform better with fuzzy overlapping while others favour crisp overlapping. We also evaluate the performance of some algorithms that recover the belonging coefficients when the overlap is fuzzy. Finally, we investigate whether real networks contain fuzzy or crisp overlapping.