Computational assessment of the discrete breathers (also known as intrinsic localised modes) is performed in nickel and palladium hydrides with an even stoichiometry by means of molecular dynamics simulations. The breathers consisting of hydrogen and metallic atoms were excited following the experience obtained earlier by modelling the breathers in pure metallic systems. Stable breathers were only found in the nickel hydride system and only for the hydrogen atoms oscillating along 〈1 0 0〉 and 〈1 1 1〉 polarization axes. At this, two types of the stable breathers involving single oscillating hydrogen and a pair of hydrogen atoms beating in antiphase mode were discovered. Analysis of the breather characteristics reveals that its frequency is located in the phonon gap or lying in the optical phonon band of phonon spectrum near the upper boundary. Analysis of the movement of atoms constituting the breather was performed to understand the mechanism that enables the breather stabilization and long-term oscillation without dissipation its energy to the surrounding atoms. It has been demonstrated that, while in palladium hydride, the dissipation of the intrinsic breather energy due to hydrogen-hydrogen attractive interaction occurs, the stable oscillation in the nickel hydride system is ensured by the negligibly weak hydrogen-hydrogen interaction acting within a distance of the breather oscillation amplitude. Thus, our analysis provides an explanation for the existence of the long-living stable breathers in metallic hydride systems. Finally, the high energy oscillating states of hydrogen atoms have been observed for the NiH and PdH lattices at finite temperatures which can be interpreted as a fingerprint of the finite-temperature analogues of the discrete breathers.