I've seen the opposite happens, the closer to the deadline the worse the estimates. At which point the entire team is in constant crunch mode and getting tired.

I've seen the opposite happens, the closer to the deadline the worse the estimates. At which point the entire team is in constant crunch mode and getting tired. This causes even more defects, which cause even worse estimates... One project I worked on last year saw four different authentication systems implemented. The last one was built on the leftovers of the scaffoldings of the previous three. We couldn't get rid of it because many other systems had been tightly integrated into authentication at various points in time. I would say the quality of the estimates aren't a function of the project's completion but instead a function of the quality and simplicity of the codebase.