Least-squares (LS) estimation is a standard tool for the optimal processing of geodetic data. In the framework of global gravity field modelling, for example, such methods are extensively applied for the determination of geoid solutions from CHAMP and GRACE data via the estimation of a large set of spherical harmonic coefficients. Frequently, in geodetic applications additional nuisance parameters need to be included in the least-squares adjustment procedure to account for external biases and disturbances that have affected the available measurements. The objective of this paper is to expose a trade-off which exists in the LS inversion of linear models that are augmented by additional parameters in the presence of unknown systematic errors in the input data. Specifically, it is shown that if a linear model of full rank is extended by a scalar parameter to account for a common bias in the data, the LS estimation accuracy of all the other model parameters automatically worsens. Some simple numerical examples are also given to demonstrate the significance of this accuracy degradation in the geodetic practice.