Bill's question seems to generate a lot of nonpositive responses, e.g., embarrassing stories of lack of knowledge.This is a great question! I give two example from my own research that almost qualify--- I used knowledge from outside of my tiny area, though I would not call the results significant. Readers- do what I do, put aside the question of significance (a topic for another post perhaps) and just comment on when knowing stuff from a different area or areas helped you in research.
How about a slightly more positive question: are there examples where someone KNOWING something from slightly outside their tiny area of expertise led to significant results?
- Knowing omega-automata and Hilbert's tenth problem was the key to the paper Learning via Queries which was co-authored with Carl Smith and appeared in FOCS 1988 and JACM 1992. It was the first in a series of papers that added queries to the Inductive Inference model.
- There is a trick in recursive graph theory where you build a graph G with two special vertices u and v so that if G is properly k-colored then u and v are the same color. This is NOT a hard trick. I used a variant of this trick to find better bounds for some poly VDW numbers. This lead to, in turn, a problem on the Maryland Math Competition, two projects for High School Students, one project for college students, and finally a paper (in progress) by Gasarch-Kruskal-Kruskal.