Sideband asymmetry between the Stokes and anti-Stokes scattering is a signature of the mechanical oscillators in the quantum regime. It has been demonstrated in several optomechanical systems, as a self-calibrated thermometry method. We present motional sideband asymmetry. measurements of a nano-optomechanical system sideband-cooled close to the ground state, probing simultaneously with red- and blue-detuned tones. Due to a Kerr-type nonlinear cavity response, the measurement can exhibit an artificially increased motional sideband asymmetry. We develop a theoretical model based on Floquet theory that accurately describes our observations. This has wide-ranging implications for schemes utilizing multiple probing or pumping tones.