  We address the question of how well the density profile of galaxy clusters
can be determined by combining strong lensing and velocity dispersion data. We
use cosmological dark matter simulations of clusters to test the reliability of
the method, producing mock catalogues of tangential and radial gravitational
arcs and simulating the radial velocity dispersion profile of the cluster
brightest central galaxy. The density profiles of the simulated clusters
closely follow the NFW form, but we find that the recovered values of the inner
slope are systematically underestimated, by about 0.4 in the mean, if the lens
is assumed to be axially symmetric. However, if the ellipticity and orientation
of the iso-contours of the cluster lensing potential are taken into account,
then the inner slopes can be recovered quite accurately for a significant
subset of the clusters whose central surface density profiles appear the most
regular. These have lensing potentials with ellipticities in the range
0.15-0.4. Further simulations projecting one cluster along many random
lines-of-sight show that, even for lower ellipticities, the central slopes are
underestimated by $\sim 10-35%$. These simulations closely mimic past
observations (see e.g. Sand et al., 2004), suggesting that existing estimates
of the central slopes may be biased towards low values. For the remaining
clusters, where the lensing potential is strongly perturbed by active merging
or by substructure, the correct determination of the inner slope requires a
more accurate model for the lens. When the halo profile is modelled by a
generalised NFW profile, we find that the inferred scale radius and
characteristic density, unlike the inner slope, are generally poorly
constrained, since there is a strong degeneracy between these two parameters.
