We discuss how locality in AdS duals to Conformal Field Theories emerge vis-a-vis bulk Effective Field Theory descriptions with a high-scale cut-off, and the emergence of a weakly-coupled Eikonal limit for high-spin CFT operators. First, we provide necessary and sufficient conditions for a CFT to have a description in terms of a local EFT in AdS. This includes a condition that the Mellin amplitudes of the CFT correlators must be well-approximated by functions that are bounded by a polynomial at infinity in Mellin space, or in other words, that the Mellin amplitudes have an effective theory-type expansion. Next, we take an analytic approach to the CFT bootstrap, studying an Eikonal-type limit of d > 2 dimensional CFT 4-pt correlators where the conformal cross ratios satisfy |u| ≪ |v| ≪ 1. We use this limit to derive exact results concerning the CFT spectrum and OPE coefficients for operators with large angular momentum. We interpret the results as a statement about super-horizon locality in AdS for very general CFTs