We have measured the dynamic structure factor of liquid neon-hydrogen mixtures (T=30.1 K) at two different H_{2} concentration levels (namely, 3.4% and 10%) making use of inelastic neutron scattering. This system has been selected since the presence of heavy Ne atoms strongly influences the self-dynamics of the H_{2} centers of mass via the formation of short-lived cages, which act both on the vibrational and the diffusive parts of the single-particle motion. After operating a standard data reduction and the subtraction of the Ne signal, experimental neutron spectra were analyzed through a generalization of the Young and Koppel model, and the H_{2} center-of-mass self-dynamic structure factor was finally extracted for the two liquid samples. Important physical quantities (namely, single-particle mean kinetic energy and self-diffusion coefficient) were estimated from the experimental data and then compared with quantum dynamical calculations, which also provided simulations of the velocity autocorrelation functions for Ne atoms and H_{2} centers of mass. The latter estimates, in the framework of the well-known Gaussian approximation, were used to simulate the H_{2} center-of-mass self-dynamic structure factor in the same kinematic range and thermodynamic conditions of the neutron scattering one. The comparison between measured and calculated spectra turned out to be qualitatively good, but some discrepancies, especially in the low-frequency part, seem to reinforce the idea of the existence of relevant non-Gaussian effects as in the case of pure hydrogen and H_{2}-D_{2} mixtures.