I'm sure there are many examples of AI doing something silly, I want to mention this one:
When 2 AI players are at war, they often attack each other's resource mining starbases. This is good, of course. But, what they don't realize is that I know that they are gonna do this, I have my constructors waiting at starbases of warring nations. So when they free up a resource by destroying a starbase, I quickly take it. I maintain peace with both nations and get away with it just fine.
I think smarter AI should see thru this sort of treachery