Safety studies of a nuclear reactor often center their interest on the probable number, of component failures in a time span of durationTo. Using the asymptotic normality of the estimator from Poisson regression, we develop approximate upper tolerance bounds for the distribution of the number of failures. Tables are given for easy computation of such bounds when the bound itself is less than 50. An example consisting of 90 failure records for nuclearreactor valve types provides illustration of the tolerance bound computations and dataanalytic techniques for validating a Poisson regression model.