  A consensus that questions the perfunctory use of the quantum adiabatic
theorem has emerged since Marzlin and Sanders [Phys. Rev. Lett. {\bf 93},
160408 (2004)] showed the existence of an inconsistency in the applicability of
the theorem. Further analysis proved that the inconsistency may arise from the
existence of resonant terms in the Hamiltonian, but recent work indicates that
the debate about the full extent of the problem remains open. Here, we first
show that key premises required in the standard demonstration of the theorem do
not hold for a dual Hamiltonian involved in the Marzlin-Sanders inconsistency.
Also, we show that two simple conditions can identify systems for which the
adiabatic approximation fails, in spite of satisfying traditional quantitative
conditions that were believed to guarantee its validity. Finally, we prove that
the inconsistency only arises for Hamiltonians that contain resonant terms
whose amplitudes go asymptotically to zero.
