We consider a dynamical formulation of network flows, whereby the network is modeled as a switched system of ordinary differential equations derived from mass conservation laws on directed graphs with a single origin-destination pair and a constant inflow at the origin. The rate of change of the density on each link of the network equals the difference between the inflow and the outflow on that link. The inflow to a link is determined by the total flow arriving to the tail node of that link and the routing policy at that tail node. The outflow from a link is modeled to depend on the current density on that link through a flow function. Every link is assumed to have finite capacity for density and the flow function is modeled to be strictly increasing up to the maximum density. A link becomes inactive when the density on it reaches the capacity. A node fails if all its outgoing links become inactive, and such node failures can propagate through the network due to rerouting of flow. We prove some properties of these dynamical networks and study the resilience of such networks under distributed routing policies with respect to perturbations that reduce link-wise flow functions. In particular, we propose an algorithm to compute upper bounds on the maximum resilience over all distributed routing policies, and discuss examples that highlight the role of cascading failures on the resilience of the network.National Science Foundation (U.S.). Office of Emerging Frontiers in Research and Innovation (ARES Grant 0735956)United States. Air Force Office of Scientific Research (Grant FA9550-09-1-0538)