Assume that one is interested in estimating an average treatment effect (ATE), equal to a weighted average of S conditional average treatment effects (CATEs). One has unbiased estimators of the CATEs. One could just average the CATE estimators, to form an unbiased estimator of the ATE. However, some CATE estimators may be less precise than others. Then, downweighting the imprecisely estimated CATEs may lead to a lower mean-squared error and/or shorter confidence intervals. This paper investigates this bias-variance trade-off, by deriving minimax linear estimators of, and confidence intervals (CI) for, the ATE, under various restrictions on the CATEs. First, I assume that the magnitude of the CATEs is bounded. Then I assume that their heterogeneity is bounded. I use my results to revisit two empirical applications, and find that minimax-linear estimators and CIs lead to small but non-negligible precision gains.