The necessary assumption that the instantaneous flow field seen by hot-wire probes with two or more sensors is uniform, i.e. that all sensors are cooled by identically the same instantaneous velocity field, is often quite erroneous in highly sheared turbulent flow, such as near the wall in a turbulent boundary layer. Intense local shear layers occur, resulting in large instantaneous velocity gradients across the sensing volume of the probe. The effects of these neglected velocity gradients on the ability of a four-sensor probe, consisting of a pair of orthogonal X-arrays, to measure the three velocity and the streamwise vorticity components is assessed. This is done by determining the synthetic response of the probe to the turbulent boundary layer database of Balint et al. (1991), in which all the velocity gradients are known. The effects of neglecting the binormal components of velocity which cool each sensor are also assessed, when the probe is treated as two uncoupled X-arrays. A small improvement to the probe's coupled X-array response is found when an estimate of the mean wall-normal velocity gradient is incorporated in the response equations.