This paper is concerned with responses of linear systems to non-Gaussian random excitation and with the measurement of the departure of the responses from Gaussian behavior. First, we show the classical Rosenblatt result and its nonapplicability to the most popular practical systems described by differential equations of first and second order. Then, using a simple measure of departure from normality (the asymmetry and excess coefficients) and performing numerical calculations, we give quantitative information about the effect of system parameters and the radius of correlation of the excitation process on the distance from normality.