For a wide class of chemical reaction networks, including all those governed by detailed balanced mass-action kinetics, we examine the robustness of equilibrium species concentrations against fluctuations in the overall reactant supply. In particular, we present lower bounds on the individual species-concentration sensitivities that derive from reaction network structure alone, independent of kinetic parameters or even of the particular equilibrium state at which sensitivities are calculated. These bounds suggest that, in the class of reaction networks considered here, very high robustness (i.e., very low sensitivities) should be expected only when the various molecules are constructed from a large number of distinct elemental building blocks that appear in high multiplicity or that combine gregariously. This situation is often encountered in biology.