But the clock doesn't START until the next day, that's the beauty of this strategy. You'd have tons of time for teams to talk trade and maybe pick up an extra pick. If nothing good comes then you just make your pick.
That said, I bet the league already has something in place that says the first round must be concluded Thursday, and no one is allowed to pass over night. If they don't they might after someone inside reads this thread, LOL. Because it's not just the last guy, the Pats could do it with nearly the same effectiveness, as could the 49ers and several other late drafting teams.