Reported here is a detailed study of the vibrational relaxation of C2H2 and C2HD upon scattering from LiF(001). While direct (specular) scattering shows no vibrational quenching for either molecule, the residence times associated with trapping-desorption are long enough to give significant quenching. By studying this quenching as a function of surface temperature, thereby varying the residence times, we show that vibrational relaxation proceeds via multiple steps. The differences between C2H2 and C2HD quenching can be understood in terms of the presence of a Fermi resonance in the former, which enhances the relaxation rates associated with the early steps. A simple kinetic model is used to fit the data, confirming that vibrational quenching proceeds via a cascading mechanism, which is likely to be quite general for polyatomic molecules.