Recent results regarding games with congestion-averse utilities (or, congestion-averse games—CAGs) have shown they possess some very desirable properties. Specifically, they have pure strategy Nash equilibria, which may be found by a polynomial time algorithm. However, these results were accompanied by a very limiting assumption that each player is capable of using any subset of its available set of resources. This is often unrealistic—for example, resources may have complementarities between them such that a minimal number of resources is required for any to be useful. To remove this restriction, in this paper we prove the existence and tractability of a pure strategy equilibrium for a much more general setting where each player is given a matroid over the set of resources, along with the (upper and lower) bounds on the size of a subset of resources to be selected, and its strategy space consists of all elements of this matroid that fit in the given size range. (This, in particular, includes the possibility of having a full matroid, or having a set of bases of a matroid.) Moreover, we show that if a player strategy space in a given CAG does not satisfy these matroid properties, then a pure strategy equilibrium need not exist, and in fact the determination of whether or not a game has a pure strategy Nash equilibrium is NP-complete. We further prove analogous results for each of the congestion-averse conditions on utility functions, thus showing that current assumptions on strategy and utility structures in this model cannot be relaxed anymore.