The numerical investigation of many-body quantum systems usually requires different kinds of physical approximations. The error which is made by these approximations is difficult to estimate and remains unknown in most cases. We examine an upper bound on expectation values of quantum subsystems, which enables the estimation of the maximum error that is made by physical approximations outside the subsystem. This is of special interest for perturbation theory, where the bath is commonly approximated with simplified interactions. A recently realized all-spin-based atomic-scale logic device, consisting of iron atoms and cobalt islands placed on a copper substrate, serves as a specific example for an application of the bound. Strength and weakness of these methods are critically discussed and we provide a quantitative answer to the old question in which cases a small quantum system can be used instead of a large one.