David vs. Goliath

Engineer: There's no way I can finish this in 3 weeks. I need 4.
Manager: If I extend your time-frame now, it would push everything else out.
Engineer: I just don't see how it is possible.
Manager: It's just one week. Have you challenged yourself to get it done in 3 weeks?
At this point, I believe the engineer has two good responses available to him. He either says, "No, I have not" or "I've done the calculations and they clearly indicate a minimum 4 weeks." If the engineer has not challenged himself, this is the time to do so. Otherwise, the manager should recognize the integrity of the engineer's calculations and seek viable alternatives (without firing the engineer).
All in all, I think it takes a person of character to stare a challenge in the face. Just as David slew Goliath, sometimes we have to change something about ourselves in order to achieve something important to us. If we put forth 100% effort and fail, it doesn't mean that we didn't want to succeed. Maybe we just weren't capable of it, but at least we can work on improving our skills and abilities so we'll be better prepared next time. The real tragedy is when we merely stand with our hands on our sides, shake our heads, and walk away, because it reflects how little we valued the goal in the first place.