Formation and breaking of metal-hydrogen bonds are central to many important catalytic processes such as transition-metal catalyzed ammonia synthesis, hydrogenation reactions, and water splitting, and thus, they require an adequate theoretical description. We studied a data set of all 30 M-H and 30 M+-H bonds of the 3d, 4d, and 5d transition series; 50 of these systems have experimentally known bond dissociation enthalpies (BDE). To probe both the limit of low and high coordination number, we also studied a data set of 19 MLnH complexes. The BDEs were computed using Hartree-Fock (HF), MP2, CCSD, CCSD(T), and 10 diverse density functionals including local, GGA, hybrid GGA, meta hybrid, range-separated, and double hybrids. Our ten most important findings are as follows: (1) HF fails completely to describe the metal hydrogen bond due to its lack of static correlation; (2) this makes post-HF methods such as MP2 and even CCSD(T) perform worse than many density functionals; (3) DFT requires much more HF exchange (similar to 35% on average) to describe the pure M-H bonds than to describe other metal ligand bonds (0-20%); (4) we design a test to determine if self-interaction error (SIE) is important by correlating DFT errors against a one-electron SIE metric; (5) we show that SIE correlates directly with the DFT errors and thus causes most of the problem; (6) HF-DFT cannot handle these systems because the HF method is too pathological already at the density level; (7) instead, we define and apply a simple metric of electronic abnormality as the difference in PBE energy computed at the self-consistent PBEO and SVWN densities, and this metric gives appropriate spread and effectively captures density-derived errors; (8) the low electronegativity of the metal enforces a diffuse hydride-like electron density, which make the metal hydrides primary examples of many-electron systems exhibiting SIE already at equilibrium geometries; (9) in the coordinatively saturated MLnH systems, much less HF exchange is required; i.e., the HF exchange requirements vary drastically with coordination number. Accordingly, DFT is unbalanced for any catalytic process involving both M-H and M-L bonds and changing coordination numbers; (10) importantly, the range-separated and double-hybrid functionals CAM-B3LYP and B2PLYP alone perform well for both M-H and M-L systems and in both limits of low and high coordination number, and at least as well as CCSD(T). This lends hope to a balanced treatment of computational chemistry for all types of M-L bonds at variable coordination number, as required for real catalytic reactions.