We study fully localized Bogomol’nyi-Prasad-Sommerfield brane solutions in classical supergravity using a perturbative approach to the coupled Born-Infeld or bulk supergravity system. We derive first order bulk supergravity fields for world-volume solitons corresponding to intersecting M2-branes and to a fundamental string ending on a D3-brane. One interesting feature is the appearance of certain off-diagonal metric components and the corresponding components of the gauge potentials. Making use of a supersymmetric ansatz for the exact fields, we formulate a perturbative expansion which applies to M2⊥M2(0), M5⊥M5(3) and Dp⊥Dp (p-2) intersections. We find that perturbation theory qualitatively distinguishes between certain cases: perturbation theory breaks down at second order for intersecting M2-branes and Dp-branes with p<~3 while it is well behaved, at least to this order, for the remaining cases. This indicates that the behavior of the full nonlinear intersecting Dp-brane solutions may be qualitatively different for p<~3 than for p>~4, and that fully localized asymptotically flat solutions for p<~3 may not exist. We discuss the consistency of these results with world-volume field theory properties.