Congestion games allow to model competitive resource sharing in various distributed systems. Pure Nash equilibria, that are stable outcomes of a game, could be far from being socially optimal. Our goal is to identify combinatorial structures that limit the inefficiency of equilibria. This question has been mainly investigated for congestion games defined over networks. Instead, we focus on symmetric matroid congestion games, where the strategies of every player are the bases of a given matroid. We derive new upper bounds on the Price of Anarchy (PoA) of congestion games defined over k-uniform matroids and paving matroids. Our bounds indicate that, in the presence of these combinatorial
structures, it is not possible to achieve the worst-case PoA of general congestion games.