Identifying the determinants of cumulative cultural evolution is a key issue in the interdisciplinary field of cultural evolution. A widely held view is that large and well-connected social networks facilitate cumulative cultural evolution because they promote the spread of useful cultural traits and prevent the loss of cultural knowledge through factors such as drift. This view stems from models that focus on the transmission of cultural information, without considering how new cultural traits actually arise. In this paper, we review the literature from various fields that suggest that, under some circumstances, increased connectedness can decrease cultural diversity and reduce innovation rates. Incorporating this idea into an agent-based model, we explore the effect of population fragmentation on cumulative culture and show that, for a given population size, there exists an intermediate level of population fragmentation that maximizes the rate of cumulative cultural evolution. This result is explained by the fact that fully connected, non-fragmented, populations are able to maintain complex cultural traits but produce insufficient variation and so lack the cultural diversity required to produce highly complex cultural traits. Conversely, highly fragmented populations produce a variety of cultural traits but cannot maintain complex ones. In populations with intermediate levels of fragmentation, cultural loss and cultural diversity are balanced in a way that maximizes cultural complexity. Our results suggest that population structure needs to be taken into account when investigating the relationship between demography and cumulative culture.This article is part of the theme issue ‘Bridging cultural gaps: interdisciplinary studies in human cultural evolution’.