Opportunities exist to apply nonlinear filtering to model-based prognostics in order to provide a systematic way of dealing with the propagation of system damage at some future time, whenever imprecise diagnostic information is obtained. Central to the prognostics problem is the ability to properly capture and manage uncertainties when predicting remaining useful life of a particular component of interest. The goal of this paper is to present a foundation for prediction and filtering of the failure process using nonlinear prognostic models and exact (finite-dimensional) filters. Specifically, we consider the use of nonlinear filters to represent the uncertainty distributions exactly for certain classes of nonlinear systems, given a statistically-representative process model of remaining useful life. One such filter, known as the Beneˇs filter, is derived in this paper for a certain class of prognostic process model. The filter is applied to crack growth data and is shown to perform reasonably well in the context of the 1-D hyperbolic model. Although directly applicable to certain prognostic systems, the techniques described provide a theoretical foundation for approximate but less model-restrictive techniques for dynamic model-based prognostics such as particle filtering.