We present an experimental study of the nonlinear absorption of graphene-on-silicon waveguides. The wavelength dependence of absorption saturation from a narrow linewidth optical pump is measured. Spectral hole burning (SHB) introduces a decrease in the probe absorption near the pump wavelength, which may be distinguished from the free carrier absorption (FCA) by using the time dependence of the buildup of the free carrier population. Beyond SHB bandwidth, only FCA dominates the waveguide loss. The spectral bandwidth of absorption bleaching from SHB is measured to have a full width at half-maximum of ~12 meV (~20 nm) at 16.8 dBm pump power.