I once had a manager hand me a project brief and ask me how quickly I thought I could complete it. I was managing my own workload (it was a bad situation), but it was a very small project and I felt that I had time to put everything else on hold and focus on it. So, I said that I might be able to get it done in four days, but I wouldn't commit to less than a week just to be sure.
The manger started off on this half-threatening, half-disappointed rant about how the project had a deadline set in stone (in four days' time), and how the head of the company had committed to it in public (which in hindsight was absolute rot). I was young and nervous, but fortunately for me every project brief had a timeline of who had seen it, and more importantly, when they had received it. I noticed that this brief had originated over three months prior, and had been sitting on this manager's desk for almost a month. I was the first developer in the chain. That gave me the guts to say that my estimate was firm, and that if anyone actually came down the ladder looking for heads to set rolling (one of the manager's threats), they could come to me and I would explain.
In the end nothing ever came of it because I managed to get the job done in three days. They tried to put the screws to me over that small of a project.