This paper investigates the accuracy of the linear model estimate that forms a part of an overall Hammerstein model structure. A key finding here is that the process of estimating a non-linear element can have a dramatic effect on the associated estimate of the linear component. This effect is not explained simply by way of considering how the input spectrum is changed by the non-linearity. Instead, it arises that the estimate variability may be dominated by a term that depends on the frequency response of the linear system itself. This has implications for experiment design for Hammerstein system estimation.