The performance of current density functionals is analyzed in detail for the electric field gradients (EFG) of hydrogen chloride and copper chloride by comparison with ab initio methods and available experimental data. The range of density functionals applied shows good agreement with coupled cluster H and Cl field gradients for HCl, as has been demonstrated previously for other main-group element containing compounds. However, the performance of most density functionals is very poor for the Cu EFG in CuCl (EFG for Cu -0.44 a.u. at the coupled-cluster singles and doubles with perturbative triples [CCSD(T)] level, compared to, e.g., +0.54 a.u. at the B-LYP level). Only the "half-and-half" hybrid functionals give field gradients with the correct sign. The reason for the poor performance of the density functional theory is analyzed in detail comparing density functional with ab initio total electronic densities rho(r). Due to the conservation of the number of particles, a change in the valence part of the electron density can lead to changes in the core part of the density. Errors in valence electronic properties like the dipole moment and in core properties like the Cu and Cl EFGs may therefore be connected. In fact the errors in both properties show a distinct linear relationship, indicating that if the dipole moment is correctly described by density functionals, the Cu and Cl EFGs may be accurate as well. Furthermore, at the atomic level, electric field gradients are described with reasonable accuracy by current density functionals as calculations for the Cu P-2 excited state and the Cu2+ D-2 ground state show. A comparison between the different density functionals shows that the incorrect behavior of the electronic density appears to be mainly due to defects in the exchange part of the functional.