This has less to do with "due dates" and more to do with "being busy." I hear this excuse EVERY DAY:
"Sam, can we talk setting priorities re: looming business risks?" "I'm busy now, let's do later"
"Sue, have you started on the Job Description updates?" "No, I'm busy with other priorities."
"Kathy, we need to go over this Plan of Correction?" "I'm busy right now."
Where I work push back is almost necessary due to unrealistic goals, spending about 50-70% of your time in meetings, change management which can take 6 months to get 1 change request approved and a lack of resources.
You guys clearly don't work for the same organisation as I do.
Having to push back on due dates here stems from the original brief not being correct, the resource you had assigned to the job being ambushed by your own version of PHB and your own time being spent in meetings explaining how far behind you are with the project because of all the meetings you have to attend to explain how far behind you are with the project, usually repeating the same thing to several levels of hierarchy at different times, rather than getting all stake holders in the same room at the same time for one almighty explanation.
I think I am reading the situation a little differently from most other people. It seems he is being told due dates without being consulted first. I have no sympathy for people who commit to delivery dates and miss them. However, if this was me, I would not be happy being told the only reasons I might have for not meeting someone else's idea of due dates was my incompetence or laziness.