The influence of moving boundaries on the stability of quantum Hamiltonian systems, in particular on the dynamics of quantum versions of the classical Pustilnikov model, is investigated (the latter consists of a masspoint bouncing above an oscillating plate under the influence of constant gravity.) It is shown that, in contrast to the classical Pustilnikov model, generic time-periodic boundary conditions (including the Dirichlet condition) on the quantum models do not allow unlimited energy gain (``speeding up'') of these systems.