I'll answer the question with respect to the Green's/Stokes' theorem special case. The summary is that, from a pedagogical point of view, little is gained by passing directly to the general case, and much is potentially lost (students for example).
If by Stokes' theorem is meant the version relating surface integrals and line integrals, this theorem is substantially harder to teach than its planar special case usually known as Green's theorem. Green's theorem relates an ordinary double integral over a region with a line integrals over the region's boundary. The version for surfaces requires defining surface integrals, and this introduces substantial new difficulties. In particular students have a lot of trouble with parameterizing surfaces, calculating normal vectors, and understanding orientations of surfaces. With Green's theorem only the orientation issue is present as such, and it is more easily visualized in the planar context, where it already is difficult for many students. In any case, objects in three dimensions are almost always harder to understand than their counterparts in two dimensions, if only because it is easier to draw pictures of planar objects.
If by Stokes' theorem one means the full blown theorem in terms of differential forms, then the difficulty, both technically and conceptually, is substantially greater. Moreover, the theorem is usually formulated in terms of differential forms (why stop there? one could go on to currents and just teach out of Federer ...). Understanding differential forms requires understanding duality of vector spaces, a topic often omitted from introductory linear algebra courses, often omitted completely from instruction given to physicists and engineers, and itself requiring some effort to assimilate properly. Many professional engineers never learn that the curl of a vector field is the vector dual to the exterior differential of the one-form dual to the vector field; physicists mostly can interpret such a phrase, but even applied physicists (as opposed to high energy theoreticians or general relativists) don't usually think that way, and don't write Maxwell's equations using forms (obviously theoreticians do).
While in fact the general theorem is not much more difficult to prove rigorously than are its various special cases (although to do so in full generality one has to deal with triangulations, or partitions of unity, or something along those lines) doing so obscures the essential ideas, which are just as well understood in proving special cases such as Green's or Gauss's (the divergence) theorem. In fact, once one understands the proof of Green's theorem, one basically understands what is needed to prove the general theorem; all the essential issues are present already.
Very few students learn top down (only rarely does one have a future Grothendieck as a first-year student). Most learn best proceeding from the special to the general. One pares away as many inessential aspects as one can, to focus the student's attention on the core issues.