Elastic scattering of C-12, N-14, and O-16 ions by aluminum, silicon, and titanium has been investigated below 34 MeV at a backscattering angle of 170 degrees. The excitation functions at energies near the Coulomb barrier and non-Rutherford threshold energies above which the Rutherford cross sections become invalid, are presented for direct scattering and the kinematically reversed reactions. Sharply decreasing cross sections above the threshold energies, but no broad resonance-like structures, were observed. The non-Rutherford threshold energies are clearly higher than the values predicted by a theoretical model based on nuclear perturbation and optical model calculations recently presented in the literature. A new unpublished version of the model gives better agreement.