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Algorithms for “community” or cohesive subgraph detection are frequently used to analyze network representations of complex systems, and constitute an active area of methodological research. Algorithms for overlapping community detection are of particular interest, because it is often unrealistic to form strict partitions of natural systems, with at least some nodes participating in multiple clusters. However, the literature on overlapping communities does not distinguish different sources of potential overlap. This paper begins by discussing randomness, bridging nodes, and superimposed relational structures as three distinct sources of overlapping communities. It then focuses on the third source (neglected in the literature), by constructing test graphs that merge two distinct relational structures and comparing the communities identified by eight partition and overlapping community algorithms. Results show that most algorithms focus on the same source structure in each graph, ignoring the other, although a few algorithms find combined or nested communities from both structures.