Given that climate projections are inherently probabilistic, it is essential for climate models to accurately represent the probability density functions (PDFs) of daily anomalies in order to reliably project the changing risks of extreme weather. Given that such PDFs are generally skewed and heavy-tailed, a minimum requirement is to represent not only the mean and variance (the first and second statistical moments) but also the skewness and kurtosis (third and fourth moments) of the PDFs. But how good are current models in this regard, and is increasing model resolution the only way to make them better? We have addressed this issue by examining the first four statistical moments of daily anomalies in integrations of the ECMWF atmospheric model run at progressively higher resolutions from T95 to T2047 for the 1989 to 2007 period with prescribed observed boundary forcings. These simulated moments were compared with corresponding observed moments estimated from the ERA-Interim reanalysis dataset. A remarkable and unexpected conclusion from these intercomparisons was that the "difficult" moments of skewness and kurtosis were actually much better represented in the simulations than the mean and variance. In other words, at all resolutions the model had less trouble capturing the distinctive non-Gaussian shapes of the observed PDFs than their location (mean) and dispersion (variance). Indeed, increasing model resolution led to a unmistakable spurious increase in the daily variance. These results raise important questions about the necessity or even the desirability of continued increases in climate model resolution for reliably predicting changes in extreme weather risks. We will argue that the T511 resolution should be sufficient for these purposes.