Very recently I and Misha Sodin had a strong incentive to learn the Ito-Nisio lemma (which, roughly speaking, says that weak convergence in probability of a series of symmetric independent random variables with values in a separable Banach space implies almost sure norm convergence to the same limit). The textbooks we could find fell into 2 categories: those that didn't present the proof at all and those presenting it on page 2xx as a combination of theorems 3.x.x, 4.x.x, 5.x.x, etc. The original paper is less than 10 pages long, essentially self-contained, and very easy to read and understand.

The moral is the same as Boris put forth: the books are there to optimize the time you need to spend to learn the whole theory. However, for every particular implication A->B the approach they usually take is something like E->F->G, G->F, (F and Q)->B; since A->E, then A->G; once we know G, we have F, so it suffices to prove that A->Q to show that A->B; we show that Q,R,S,T,U are equivalent, with the trivial implication S->Q left to the reader as an exercise; finally, we prove that A->S. So if all you need is A->B, you may be much better off reading the paper whose only purpose is to prove exactly that.