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Bounds are established for the probability of failure of fault-tolerant systems. The underlying failure and recovery process is assumed to follow a semi-Markov model in which the potential sojourn times of component failures have exponential distributions and those of system responses have general distributions. A product form of the bounds is derived from a model which provides for competing responses to component failures. Bounds are calculated in terms of integral factors which depend on component failure rates and the actual distributions of response times. Bounds are also calculated in terms of percentiles, conditional mean response times, and certain transition probabilities. Besides providing a computationally efficient method of estimating reliability from a flexible model, the bounds seem reasonably tight for a fairly wide range of cases. An example is given to illustrate calculating the bounds for a model with competing system responses.