We introduce the Generative Barankin Bound (GBB), a learned Barankin Bound, for evaluating the achievable performance in estimating the direction of arrival (DOA) of a source in non-asymptotic conditions, when the statistics of the measurement are unknown. We first learn the measurement distribution using a conditional normalizing flow (CNF) and then use it to derive the GBB. We show that the resulting learned bound approximates the analytical Barankin bound well for the case of a Gaussian signal in Gaussian noise, Then, we evaluate the GBB for cases where analytical expressions for the Barankin Bound cannot be derived. In particular, we study the effect of non-Gaussian scenarios on the threshold SNR.