+100 to this, any time I get an 'emergency request' coming in like that I flat out tell them this is not going to happen, I can't do it given the time constraint. I didn't used to do that, but I got burned too many times and it hurt my mental health scrambling to meet someone else's fabricated, and usually self-inflicted deadline. My well-being is so much more important than somebody else's lack of planning. My usual strategy for dealing with this is saying "Sorry, I can't help given the time constraint. Since it's an emergency maybe we should bring this to the attention of <relevant manager 2 levels up>?" and magically the 'emergency' goes away and I never hear about it again.