The Bruhat order hides a lot of combinatorial structure, yet two of its most natural questions remain wide open: Given an interval, how many elements does it contain? And how can we classify these intervals as posets? Recent results suggest surprisingly elegant answers for these questions, and for affine Weyl groups, we have found that a small part of the poset almost distinguishes between non-isomorphic Bruhat intervals. Perhaps ideas from machine learning could be the key to closing this gap.