We investigate the effect of frequency on the non-linear large amplitude oscillatory shear (LAOS) response of concentrated colloidal suspensions of model soft and hard spheres at various concentrations, both below and above the glass transition. We show that the anharmonic response in the stress increases with frequency for liquid-like samples but decreases with frequency for solid-like samples. We argue that for samples below the glass transition, higher frequencies involving higher maximum shear rates promote shear thinning and increase anharmonicity. On the other hand, solid-like samples deform plastically at low frequencies as they are subjected to low shear rates within the period. Higher frequencies (higher average shear rates) lead to viscous flow over a larger fraction of the period thereby decreasing anharmonic behavior. We also demonstrate that LAOS experiments in strain-controlled rheometry at moderately high frequencies (ω > 5 rad/s) have to be very carefully interpreted, due to the superharmonic instrumental resonance effects. © 2015, Springer-Verlag Berlin Heidelberg.