Heterogeneity has been studied as one of the most common explanations of the puzzle of cooperation in social dilemmas. A large number of papers have been published discussing the effects of increasing heterogeneity in structured populations of agents, where it has been established that heterogeneity may favour cooperative behaviour if it supports agents to locally coordinate their strategies. In this paper, assuming an existing model of a heterogeneous weighted network, we aim to further this analysis by exploring the relationship (if any) between heterogeneity and cooperation. We adopt a weighted network which is fully populated by agents playing both the Prisoner's Dilemma or the Optional Prisoner's Dilemma games with coevolutionary rules, i.e., not only the strategies but also the link weights evolve over time. Surprisingly, results show that the heterogeneity of link weights (states) on their own does not always promote cooperation; rather cooperation is actually favoured by the increase in the number of overlapping states and not by the heterogeneity itself. We believe that these results can guide further research towards a more accurate analysis of the role of heterogeneity in social dilemmas. (C) 2017 Elsevier B.V. All rights reserved.