While numerous studies have suggested that large natural, biological, social, and technological networks are fragile, convincing theories are still lacking to explain why natural evolution and human design have failed to optimize networks and avoid fragility. In this paper we provide analytical and numerical evidence that a tradeoff exists in networks with linear dynamics, according to which general measures of robustness and performance are in fact competitive features that cannot be simultaneously optimized. Our findings show that large networks can either be robust to variations of their weights and parameters, or efficient in responding to external stimuli, processing noise, or transmitting information across long distances. As illustrated in our numerical studies, this performance tradeoff seems agnostic to the specific application domain, and in fact it applies to simplified models of ecological, neuronal, and traffic networks.