The distribution of model-based estimates of equilibrium climate sensitivity has not changed substantially in more than 30 years. Efforts to narrow this distribution by weighting projections according to measures of model fidelity have so far failed, largely because climate sensitivity is independent of current measures of skill in current ensembles of models. This work presents a cautionary example showing that measures of model fidelity that are effective at narrowing the distribution of future projections (because they are systematically related to climate sensitivity in an ensemble of models) may be poor measures of the likelihood that a model will provide an accurate estimate of climate sensitivity (and thus degrade distributions of projections if they are used as weights). Furthermore, it appears unlikely that statistical tests alone can identify robust measures of likelihood. The conclusions are drawn from two ensembles: one obtained by perturbing parameters in a single climate model and a second containing the majority of the world’s climate models. The simple ensemble reproduces many aspects of the multimodel ensemble, including the distributions of skill in reproducing the present-day climatology of clouds and radiation, the distribution of climate sensitivity, and the dependence of climate sensitivity on certain cloud regimes. Weighting by error measures targeted on those regimes permits the development of tighter relationships between climate sensitivity and model error and, hence, narrower distributions of climate sensitivity in the simple ensemble. These relationships, however, do not carry into the multimodel ensemble. This suggests that model weighting based on statistical relationships alone is unfounded and perhaps that climate model errors are still large enough that model weighting is not sensible.