No, not in my experience! Both American Express and Bank of America have called me before, regarding suspicious activity on my cards. (Which turned out to be fraud.) I'm glad they did! That immediate attention sure beats waiting for sending mail, me receiving it, then acting on it. By calling me as soon as their system flagged suspicious activity, and I confirmed it was not me, they canceled the cards and stopped the fraud in its tracks!
Of course, I was absolutely sure who I was talking to. I mean I knew they weren't spoofers/scammers/spammers/other lowlifes like that. Just because caller ID says "American Express" and has a number I know belongs to AmEx, doesn't mean it *is* AmEx, you know? Any spoofer can do that. So it's important to be sure who you're talking to and, if in doubt, hang up and call them back at the number on your card.