Severe safety concerns in man-made systems require that some variables remain within very strict limits. Even though there is a rich literature on probabilistic and on ‘energy’ bounds, results on hard constraints are scarce. Hard constraint bounding results are summarized here with additional new results. Novel and existing bounding techniques for the outputs and the states of systems disturbed either by uncertain inputs or modelling errors are derived, using either boxes or ellipsoids as the constraint sets. The bounding results are summarized, based on both the concept of matrix measures and Schweppe's ellipsoid bounds. Bounds for large-scale systems and non-linear systems are presented, with some of the theoretical results being applied to a simplified model of a physically meaningful system, namely a steam boiler model, with encouraging results. For the class of non-linear systems represented by the boiler model, it is concluded that even though a box bound based control system may produce more conservative results, as compared to Schweppe's ellipsoid bounding approach, it can be more attractive because of certain practical implementation issues, such as the limitation on realizable actuator bandwidths. Furthermore, for a linear system with a Metzler system matrix, box bounds are the tightest that can actually be attained.