We present a framework to quantify the extent to which an approximate Hamiltonian is a suitable model for a real Hamiltonian, based on the degree of stability of the approximate constants of motion that are exact constants in the model. By observing the evolution under the real Hamiltonian of packets prepared initially as eigenstates of the model Hamiltonian, we are able to define quantitative criteria for the quality of the approximation represented by the model. Quantitative measures emerge for the concepts of "approximate constant of the motion" and "pretty good quantum number". This approach is intended for evaluating alternative starting points for perturbational and variational calculations, and for extracting physical insights from elaborate calculations of real systems. The use of the analysis is illustrated with examples of a one-dimensional Morse oscillator approximated by a harmonic oscillator and by another Morse oscillator, and then by a less trivial system, an anharmonic, nonseparable two-dimensional oscillator, specifically a Henon-Heiles potential modified with a fourth-order term to keep all states bound. The higher the angular momentum within any given band, the better the angular momentum is conserved. The square of the angular momentum is less well conserved than the angular momentum itself.