Runaway electrons, which are generated in a plasma where the induced electric field exceeds a certain critical value, can reach very high energies in the MeV range. For such energetic electrons, radiative losses will contribute significantly to the momentum space dynamics. Under certain conditions, due to radiative momentum losses, a non-monotonic feature – a ‘bump’ – can form in the runaway electron tail, creating a potential for bump-on-tail-type instabilities to arise. Here, we study the conditions for the existence of the bump. We derive an analytical threshold condition for bump appearance and give an approximate expression for the minimum energy at which the bump can appear. Numerical calculations are performed to support the analytical derivations.