We have derived non-Coulomb phase shifts from measured differential cross sections for electron scattering by the ions Na⁺, Cs⁺, N³⁺, Ar⁸⁺ and Xe⁶⁺ at energies below the inelastic threshold. Values of the scaled squared deviation between the observed and fitted differential cross sections, χ², for the best-fit phase shifts were typically in the range 3-6 per degree of freedom. Generally good agreement with experiment is obtained, except for wide-angle scattering by Ar⁸⁺ and Xe⁶⁺. Current measurements do not define phase shifts to better than ≈ 0.1 rad even in the most favourable circumstances and uncertainties can be much larger.