To gain insight into the non-abelian Born-Infeld (NBI) action, we study coinciding D-branes wrapped on tori, slid turn on magnetic fields on their worldvolume. We then compare predictions for the spectrum of open strings stretching between these D-branes, from perturbative string theory and from the effective NBI action. Under some plausible assumptions, we find corrections to the Str-prescription for the NBI action at order FS. In the process we give a way to classify terms in the NBI action that can he written in terms of field strengths only, in terms of permutation group theory. (C) 2001 Elsevier Science B.V. All rights reserved.