Most data assimilation and satellite retrieval methods are based on optimal estimation theory, which assumes that the error covariances of the observations and of the a priori (background) information are known. The specification of background error covariance is crucial to the appropriate interpretation of radiance information from satellite sounders within a numerical weather prediction (NWP) data assimilation system. Uncertainties in the specification of error covariances are inevitable, but an improved understanding of the acceptable range of mis-specification should lead to improved impact of these observations in NWP. In this study, the sensitivity of analysis error covariance to the mis-specification of background error covariance is investigated. The linear theory for analysis error covariance is presented, both for the optimal case and for the more general suboptimal case. The theory is first applied to a simple scalar case. A ‘danger zone’ is identified, where a mis-specification of background error variance results in analysis error variances being greater than background error variances. The boundary of the ‘danger zone’ is quantified as a function of the ratio of observation error variance to background error variance. The investigation is then extended to the vector case, and specifically to the vertical eigenmodes of a forecast error covariance matrix and to the linearized error analysis for a one-dimensional variational (1D-Var) retrieval analogous to the assimilation of radiances from the Infrared Atmospheric Sounding Interferometer (IASI). In the optimal case, the assimilation of IASI data can improve the analysis of the leading eigenmodes of temperature and humidity. A suboptimal case is then investigated, by assuming one representation of forecast error covariance in the analysis whilst taking another representation as truth. The analysis is shown to cross into the ‘danger zone’ for a small number of eigenmodes. Equivalent degradation of the analysis in profile space is also shown. Copyright © 2012 British Crown copyright, the Met Office. Published by John Wiley & Sons Ltd.