The electric field gradient (EFG) tensors for Al and O atoms in corundum and for C and D(H-2) in hexadeuterobenzene (C6D6) are determined from accurate X-ray diffraction data using four independent computational strategies. Two of the strategies capitalize on the fact that the electric field gradient is the second derivative of the electrostatic potential, a simple relationship which has not been exploited previously in analyses of diffraction data. The convergence behaviour of the tenser components calculated by each strategy is examined for both systems, and the fully converged results obtained with each computational strategy are shown to be identical. It is found that the orientation of the tensors and the signs of their components are well determined from diffraction data, but the magnitudes and asymmetry of the components are not. Discrepancies in previous analyses of the corundum data are resolved, and it is shown that the origin of the EFG at hydrogen is quite different from that at heavier nuclei. Successful determination of EFGs from X-ray diffraction data appears within reach, but will require precise knowledge of atomic position and thermal parameters, more extensive data sets than are currently available, and a more flexible pseudoatom model than presently used.