The significance of dispersion contribution in the formation of strong hydrogen bonds (H-bonds) can no more be ignored. It was illustrated that less electronegative and electropositive H-bond acceptors such as S, Se, and Te are also capable of forming strong N-H center dot center dot center dot Y H-bonds, mostly due to the high polarizabilities of H-bond acceptor atoms. Herein, for the first time, we report the evidence of formation of nonconventional M-H center dot center dot center dot Y H-bonds between metal hydrides (M-H, M = Mn, Fe, Co) and chalcogen H-bond acceptors (Y = O, S, or Se). The nature and the strength of unusual M-H center dot center dot center dot Y H-bonds were revealed by several quantum chemical calculations and H-bond descriptors. The structural parameters, electron density topology, donor-acceptor natural bond orbital (NBO) interaction energies, and spectroscopic observables such as M-H stretching frequencies and H-1 chemical shifts are well-correlated to manifest the existence and strength of M-H center dot center dot center dot Y H-bonding. The M-H center dot center dot center dot Y H-bonds are dispersive in nature, and the computed H-bond energies are found to be in the range from similar to 5 to 30 kJ/mol, which can be compared to those of the conventional H-bonds such as O-H center dot center dot center dot O, N-H center dot center dot center dot O, and N-H center dot center dot center dot O=C H-bonds, etc.