If your doing a monthly achievement, do you need to complete all 4 tiers of 4 items to unlock the chest at the end or does the quality of the chest just drop when you don't get all tiers to 4?
For doing the monthly achievments in Guild Wars 2 you need to complete all 4 tiers of 4 out of 7 achievements to unlock the chest. Otherwise there is simply no chest (you receive it after finishing all 4).
You can read more about the monthly achievments on the Guild Wars 2 Official Wiki Page: Monthly
The rewards you get from the chest by the way are always the same: