We present a decomposition procedure for evaluating the reliability of a network whose elements fail independently with known probabilities. A subnetwork is “solved” by computing the probability for classes of failure patterns within which the network performance is constant. We then iterate, combining solved subnetworks until the entire network has been evaluated. The running time for each iteration depends superexponentially on the subnetwork boundary size, but for treelike or “thin” networks total time grows only linearly with total network size. Details of our decomposition scheme are presented for a wide variety of measures of performance. For measures where computing the distribution of performance is too slow we present a quicker method which finds the average performance. Finally, we show that computing network reliability is NP-hard, so that efficiency in special cases is all that can be achieved.