Cellular functions are mediated by specific molecular interactions; however, often competing nonspecific interactions can occur instead, for example, in noncoding regions of genes during transcription or in the response of cell receptors to external signals. Various functional roles have been proposed for such interactions. Motivated by these considerations, we study the time-dependent behavior of a class of discrete, stochastic models in which decoy molecules mediate nonspecific reactions that sequester activated molecules. It is shown that such nonspecific interactions can lead to a time delay in the completion of the specific reaction by the activated molecule, thus permitting discrimination between signals of different duration. We study the effect of stochastic fluctuations in a simple model of gene transcription by numerical solution of the Master Equation and find that the distribution of first passage times for the specific reaction shows surprising nonexponential (non-Debye) behavior over a range of time scales. The mathematical mechanism underlying this behavior is explained in terms of the behavior of the eigensystem of the linear operator associated with the time evolution. Our results demonstrate that stochastic sequestration can be used to enhance the specificity achieved by the well-known kinetic proofreading mechanism.