I think you're right. In fact, I'm pretty sure of it. One time, not too long ago, I was playing as the Turks. The Byzantines decided to allow a HUGE French crusade through. BUT... the Byz decided to tag along, without signing an alliance. Now, I was seriously outnumbered...Either the Byz army, or the French could have taken me out (at great cost, I had an awesome position at the top of a hill, which was my tought on the matter, I was going to inflict heavy losses & run). Well, turns out I didn't have to, because they hadn't allied before they mutually attacked me, they entered right next to each other and proceeded to do me the favor of softening each other up.

It was an incredibly costly battle, but I destroyed the crusade and caused a civil war in France. The Byz had 'bet it all on red' on that one... since I decided not to ransom their army back, they had nothing. It was a cakewalk after that.