Our Birthright game had several sessions without combat, or with very little, though it sounds like we were a bit more active than TroyXavier's.
One session consisted entirely of the party conducting archaeological exploration of a pre-Deismaar site just north of the Gorgon's Crown, and then dealing with the traps and logic puzzles of this particular site, which happened to be an abandoned lair of one of the Lost. There was no combat for the entire session, but the traps were INCREDIBLY rude. Picture a rogue, polymorphed into a spider monkey, disarming traps in a 6" wide airshaft.
Another two sessions dealt with the takeover of Alamie by a Vos barbarian horde (I believe these were from Tales of the Hero Kings); never has an invasion been so welcomed, as when the invaders, instead of attacking, throw bags of gold at the defenders. One session had the party, at the request of the half-elven mage (who ruled Tuornen, a neighboring nation that had split off from Alamie years ago) do a recon-in-force to try and rescue the princess, which wound up not having anything happen. Another was at a ball that said Vos barbarian was throwing, which wound up having lots of intriguey stuff. In fact, we almost missed the adventure, not having noticed or cared what was going on.
However, most of the rest of the sessions had some form of high-octane combat, even the archaeological dig sessions.
Brad