Never had this issue with AE specifically but have had it with other cards I possess. In fact my experience is the opposite, AE lets transactions go through to me and when a fraudulent transaction hits, the next business day I receive a new card in the mail with a letter explaining the situation. OTOH I suspect my spending habits are different than most on this forum - I probably use cash a couple of times per year and everywhere else I'll use AE exclusively or when they don't accept AE put it on one of my Visa or MCs.
That being said, you can't really blame AE here, they're just doing their job protecting you from fraud. Had they let the purchase go thru and you call them up later they may have limited recourse to get the money back (depending on the merchant). The fraud detection systems themselves use a bunch of heuristics, including the category of purchase (i.e. a purchase at a tobacco shop with my card will most certainly be declined), when it was purchased, where it was purchased, the vendor itself (i.e. high amount of fraud), etc.
-James