Fission–fusion social systems, in which members of a social community form frequently changing subgroups, occur in a number of mammalian taxa. Such systems are assumed to be a response to the costs of grouping, but evidence to support this hypothesis is limited. We use a linear programming approach to build a time budget model that predicts the upper bound on group size in order to test the hypothesis that fission–fusion social systems are the outcome of time constraints. Comparative data from 14 wild chimpanzee (Pan spp.) populations are used to derive a set of equations defining the relationship between climatic variables and time budget components, which are then used to calculate the upper limits on group size that can be maintained in different habitats. We validate the model by showing that it correctly predicts the presence/absence of chimpanzees across sub-Saharan Africa and the group sizes observed in natural populations. The model suggests that the costs of travel are limiting for chimpanzees. Chimpanzees can reduce these costs dramatically by fissioning their bonded communities into small foraging parties. If they did not, they would be unable to live in any habitats where they currently occur.