Developing novel methods for the analysis of intracellular signaling networks is essential for understanding interconnected biological processes that underlie complex human disorders. A fundamental goal of this research is to quantify the vulnerability of a signaling network to the dysfunction of one or multiple molecules, when the dysfunction is defined as an incorrect response to the input signals. In this study, we propose an efficient algorithm to identify the extreme signaling failures that can induce the most detrimental impact on the physiological function of a molecular network. The algorithm basically finds the molecules, or groups of molecules, with the maximum vulnerability, i.e., the highest probability of causing the network failure, when they are dysfunctional. We propose another algorithm that efficiently accounts for signaling feedbacks in this analysis. The algorithms are tested on two experimentally verified ERBB and T cell signaling networks. Surprisingly, results reveal that as the number of concurrently dysfunctional molecules increases, the maximum vulnerability values quickly reach to a plateau following an initial increase. This suggests the specificity of vulnerable molecule (s) involved, as a specific number of faulty molecules cause the most detrimental damage to the function of the network. Increasing a random number of simultaneously faulty molecules does not further deteriorate the function of the network. Such a group of specific molecules whose dysfunction causes the extreme signaling failures can better elucidate the molecular mechanisms underlying the pathogenesis of complex trait disorders, and can offer new insights for the development of novel therapeutics.