Much attention has been paid recently to bistability and switch-like behavior that might be resident in important biochemical reaction networks. There is, in fact, a great deal of subtlety in the relationship between the structure of a reaction network and its capacity to engender bistability. In common physicochemical settings, large classes of extremely complex networks, taken with mass action kinetics, cannot give rise to bistability no matter what values the rate constants take. On the other hand, bistable behavior can be induced in those same settings by certain very simple and classical mass action mechanisms for enzyme catalysis of a single overall reaction. We present a theorem that distinguishes between those mass action networks that might support bistable behavior and those that cannot. Moreover, we indicate how switch-like behavior results from a well-studied mechanism for the action of human dihydrofolate reductase, an important anti-cancer target.