Advantages and disadvantages of the maximum entropy method (MEM) in application to the theory of relaxation are studied. The time evolution of distributions and of associated moments must obey stringent conditions for both finite and infinite intervals. The theoretical considerations are illustrated with examples from charge-state distributions arising in beam-foil spectroscopy. The examples indicate that the possibility to include more than two moments (extension to non-Gaussian case) is severely limited (though feasible) in the static case due to nonpositive definiteness as well as stiffness of the Hessian matrices appearing in the computations. This takes place already for the finite charge-state distribution intervals. For infinite intervals, this is a severe problem as required by the Marcinkiewicz theorem, affecting characteristic functions and, hence, the description of the time evolution of distributions.