Standard methods for fixed effects meta-analysis assume that standard errors for study-specific estimates are known, not estimated. While the impact of this simplifying assumption has been shown in a few special cases, its general impact is not well understood, nor are general-purpose tools available for inference under more realistic assumptions. In this paper, we aim to elucidate the impact of using estimated standard errors in fixed effects meta-analysis, showing why it does not go away in large samples and quantifying how badly miscalibrated standard inference will be if it is ignored. We also show the important role of a particular measure of heterogeneity in this miscalibration. These developments lead to confidence intervals for fixed effects meta-analysis with improved performance for both location and scale parameters.