In this article we discuss the problem of a reliable determination of parameters of the nuclear mean-field Hamiltonians in view of obtaining high quality comparisons with the experimental single particle spectra: a. In the case of extrapolations to extreme (exotic) isospin nuclei as well as b. In terms of predictions of the energy levels that have not been so far measured also in 'non-exotic' nuclei. We formulate our discussion in terms of the Inverse Problem, a quickly growing sub-field of Applied Mathematics, and introduce a number of mathematical properties which are illustrated in Ref.(1) We focus on those mechanisms that may significantly limit the predictive power of the resulting Hamiltonians.