We introduce a model of squeezing and noise reduction in the Kerr effect that accounts for noise in all quadratures of the driving field. Consequently we show that Kerr squeezing is much more sensitive to driving noise than squeezing produced by second harmonic generation (SHG). We experimentally demonstrate this sensitivity using a nonlinear system that tunes between strong classical SHG and Kerr behaviours. Whilst the system experiences strong squeezing in the SHG limit, it experiences no squeezing in the Kerr limit, although it does experience strong classical noise reduction, or `classical squeezing'.