  The non equilibrium Green function formalism is today the standard
computational method for describing elastic transport in molecular devices.
This can be extended to include inelastic scattering by the so called
self-consistent Born approximation (SCBA), where the interaction of the
electrons with the vibrations of the molecule is assumed to be weak and it is
treated perturbatively. The validity of such an assumption and therefore of the
SCBA is difficult to establish with certainty. In this work we explore the
limitations of the SCBA by using a simple tight-binding model with the
electron-phonon coupling strength $\rm{\alpha}$ chosen as a free parameter. As
model devices we consider Au mono-atomic chains and a $\rm{H_2}$ molecule
sandwiched between Pt electrodes. In both cases our self-consistent
calculations demonstrate a breakdown of the SCBA for large $\rm{\alpha}$ and we
identify a weak and strong coupling regime. For weak coupling our SCBA results
compare closely with those obtained with exact scattering theory. However in
the strong coupling regime large deviations are found. In particular we
demonstrate that there is a critical coupling strength, characteristic of the
materials system, beyond which multiple self-consistent solutions can be found
depending on the initial conditions in the simulation. We attribute these
features to the breakdown of the perturbative expansion leading to the SCBA.
