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 Navarro, Frenk & White (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 isocontours 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 ∼10–35 per cent. These simulations closely mimic past observations, 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 generalized 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.