In applications of linear regression analysis, the unknown error covariance matrix has to be somehow estimated. This can lead to biased estimates of the covariance matrix of the regression coefficients. Since such bias is difficult to eliminate completely, its sensitivity to alternative estimates of error covariances is studied by Watson, Theil, Malinvaud, and others with the help of bounds on the bias derived under certain assumptions. This paper gives similar bounds under less restrictive assumptions, and illustrates them in the context of heteroscedasticity and autocorrelation problems. In particular, for the first order error autocorrelation coefficient of p the upper bound on proportionate bias is shown to be reasonably approximated by (1 + p)/(l - p) - 1.