  Gravitational lensing shear has the potential to be the most powerful tool
for constraining the nature of dark energy. However, accurate measurement of
galaxy shear is crucial and has been shown to be non-trivial by the Shear
TEsting Programme. Here we demonstrate a fundamental limit to the accuracy
achievable by model-fitting techniques, if oversimplistic models are used. We
show that even if galaxies have elliptical isophotes, model-fitting methods
which assume elliptical isophotes can have significant biases if they use the
wrong profile. We use noise-free simulations to show that on allowing
sufficient flexibility in the profile the biases can be made negligible. This
is no longer the case if elliptical isophote models are used to fit galaxies
made up of a bulge plus a disk, if these two components have different
ellipticities. The limiting accuracy is dependent on the galaxy shape but we
find the most significant biases for simple spiral-like galaxies. The
implications for a given cosmic shear survey will depend on the actual
distribution of galaxy morphologies in the universe, taking into account the
survey selection function and the point spread function. However our results
suggest that the impact on cosmic shear results from current and near future
surveys may be negligible. Meanwhile, these results should encourage the
development of existing approaches which are less sensitive to morphology, as
well as methods which use priors on galaxy shapes learnt from deep surveys.
