Network data can be conveniently modeled as a graph signal, where data values are assigned to the nodes of a graph describing the underlying network topology. Successful learning from network data requires methods that effectively exploit this graph structure. Graph neural networks (GNNs) provide one such method and have exhibited promising performance on a wide range of problems. Understanding why GNNs work is of paramount importance, particularly in applications involving physical networks. We focus on the property of discriminability and establish conditions under which the inclusion of pointwise nonlinearities to a stable graph filter bank leads to an increased discriminative capacity for high-eigenvalue content. We define a notion of discriminability tied to the stability of the architecture, show that GNNs are at least as discriminative as linear graph filter banks, and characterize the signals that cannot be discriminated by either.