We introduce a graceful approach to probabilistic inference called bounded conditioning. Bounded conditioning monotonically refines the bounds on posterior probabilities in a belief network with computation, and converges on final probabilities of interest with the allocation of a complete resource fraction. The approach allows a reasoner to exchange arbitrary quantities of computational resource for incremental gains in inference quality. As such, bounded conditioning holds promise as a useful inference technique for reasoning under the general conditions of uncertain and varying reasoning resouces. The algorithm solves a probabilistic bounding problem in compex belief networks by breaking the problem into a set of mutually exclusive, tractable subproblems and ordering their solution by the expected effect that each subproblem will have on the final answer. We introduce the algorithm discuss its characterization, and present its performance on several belief networks, including a complex model for reasoning about problems in intensive- care medicine.