Recent advances in technology have made it possible to build surveillance systems using many low-cost sensor nodes with limited computation and communication capabilities. Due to a potentially large number of nodes deployed, node failures are inevitable and can render a surveillance system that has degraded detection performances. The exposure metric has been proposed earlier to assess the quality of a surveillance system based on the detection performances. In this paper, we characterize the vulnerability of a system in terms of its exposure with respect to the number of faulty nodes and their combinations. Specifically, we assess the exposure of a surveillance network subject to a given number of faulty nodes, and identify the worst-case fault combination for both an idling target and a traversing target. For an idling target, the worst-case fault combination and exposure is analytically identified. For a traversing target, a genetic algorithm based approach is proposed to derive a near worst-case fault combination, and extensive simulation results are presented to show the effectiveness of the algorithm.