I think there's a few possibilities about what happened with Posey.
1) Ainge didn't think he was the worth the MLE to begin with, and simply low-balled him in years knowing he'd sign elsewhere. This clears him from public backlash for not bringing back a popular player.
2) Luxury tax issues. The C's are already pretty far over the luxury tax. While ownership may have allowed him to use it this, they might have still limited him to a certain budget each year. In other words, while Ainge can sign a player to a 5 year MLE deal every year under NBA rules, ownership may not have allowed him to do that. Thus, Posey could limit what he could do next year and the year after due to ownership policies.
3) Maybe Ainge is planning something crazy in 2-3 years. I don't see exactly how that's possible given the state of Pierce and Garnett's deals plus Rondo's extension. Plus, getting cap room would also be contingent on holding onto Ray Allen, renouncing his rights, then counting on the fact we can sign a free agent, which would be very risky business.
Anyway, except for maybe option 2, I don't think they're very good reasons to not sign Posey UNLESS he believes the Big Three's window of opportunity is closer to 4-5 years than 2-3 AND he believes that he can land someone at the MLE next year that will make us forget about James Posey.