I don't think it's really a question of temperature. It depends what process you're looking at. The BO approximation works well for equilibrium structures and properties but breaks down if you want to describe electron transfer. 

(Also note that for a single H atom the problem is separable into center of mass and relative coordinates - so there is no BO approximation really.)

   Dear Thomas,

   You are right, the proton cannot be considered to be fixed. On the other hand the H atoms don't form a crystal lattice and therefore it is not useful to think in the Born-Oppenheimer approach of the Hamiltonian.