We consider network design games with weighted players and uniform edge capacities and study their Nash equilibria. In these games, each player has to choose a path from her source to her sink through a network subject to the constraint that the total weight of all players using an edge within their chosen path does not exceed the capacity of the edge. The fixed cost of each edge that is used by some player is shared among the players using the edge by charging each player a fraction of the edge's cost equal to the ratio of her weight to the total weight of all players using the edge. We show that there exist instances of capacitated network design games with weighted players and uniform capacities that do not admit a Nash equilibrium even in the case that all players share the same source and sink. Moreover, we show that it is strongly ‐hard to decide whether a given instance admits a Nash equilibrium even if a feasible solution for the underlying network design problem is guaranteed to exist. In contrast, we prove that, for series‐parallel graphs, there always exists a Nash equilibrium whose total cost equals the cost of an optimal solution of the corresponding network design problem and provide an (exponential‐time) algorithm to compute this equilibrium. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(2), 141–158 2016