We propose an explanation for the anomalous compressibility maximum in amorphous silica based on rigidity arguments. The model considers the fact that a network structure will be rigidly compressed in the high-pressure limit, and rigidly taut in the negative pressure limit, but flexible and hence softer at intermediate pressures. We validate the plausibility of this explanation by the analysis of molecular dynamics simulations. In fact this model is quite general, and will apply to any network solid, crystalline or amorphous; there are experimental indications that support this prediction. In contrast to other ideas concerning the compressibility maximum in amorphous silica, the model presented here does not invoke the existence of polyamorphic phase transitions in the glass phase.