Too many managers figure the deadline to be ten minutes before a project is due, forgetting that it takes time for admins and office pro's to put final touches on them. Get managers used to the phrase "administrative deadline". By using that term for setting deadlines for your role in project management, the concept will become more concrete in your mind. For example, use phrases like,
If the drop-dead deadline for this is (date) the administrative deadline needs to be (date).
Use the term administrative deadlines until you hear others use it, too. That's your signal that your part in the success of projects is acknowledged and understood.