Bah, your estimate is meaningless anyway, since someone five levels above you in the org chart has already promised the customer that feature X would be availab

Bah, your estimate is meaningless anyway, since someone five levels above you in the org chart has already promised the customer that feature X would be available in 3 weeks. And that promise was made last Tuesday, so by the time it gets to you, there are only two weeks remaining until the deadline. You'll know this is the case when you say "at least a month", and the PM looks at you like you just shot his dog. "The answer I need is two weeks."