How accurate are predictions of climate change from a model which is biased against contemporary observations? If a model bias can be thought of as a state-independent linear offset, then the signal of climate change derived from a biased climate model should not be affected substantially by that model's bias. By contrast, if the processes which cause model bias are highly nonlinear, we could expect the accuracy of the climate change signal to degrade with increasing bias. Since we do not yet know the late 21st Century climate change signal, we cannot say at this stage which of these two paradigms describes best the role of model bias in studies of climate change. We therefore study this question using time-slice projections from a global climate model run at two resolutions - a resolution typical of contemporary climate models and a resolution typical of contemporary numerical weather prediction – and treat the high-resolution model as a surrogate of truth, for both 20th and 21st Century climate. We find that magnitude of the regionally varying model bias is a partial predictor of the accuracy of the regional climate change signal for both wind and precipitation. This relationship is particularly apparent for the 850 mb wind climate change signal. Our analysis lends some support to efforts to weight multi-model ensembles of climate change according to 20th Century bias, though note that the optimal weighting appears to be a nonlinear function of bias.