Given a complex of R-modules M, one can construct a variety that contains homological information about M called the cohomological support variety V_R(M). These have various homological applications – for example, Pollitz showed they can be used to characterize when R is a complete intersection. In this talk, we will discuss the following realizability question: given an appropriately chosen variety V, when can V be realized as V_R(M) for some M? To study this question, we give bounds on the dimension of such varieties. This is joint work with Ben Briggs and Josh Pollitz.