Posey's deal was extreme for a role player and I think not matching it was justified even without an obvious replacement available. It was only one year into the Big Three era and though the window was small to begin with, tying up 6-7 million in cap space for four years would have limited moves down the road. Sure it's easy to look back and criticize the lack of a successful replacement but I think you can judge the transaction on its own. It's also possible that even with the re-signing his play would have deteriorated and the team still would not have been able to repeat.
Ainge's subsequent signings were almost all bad. I wish also that Pierce and Garnett would have taken more significant discounts considering their age, closer to the average annual salary Duncan, Ginobili, and Parker settled for. I mean, just look at their salaries during the run:
Garnett:
2007-08 Boston Celtics NBA $23,750,000
2008-09 Boston Celtics NBA $24,751,934
2009-10 Boston Celtics NBA $16,400,000
2010-11 Boston Celtics NBA $18,800,000
2011-12 Boston Celtics NBA $21,247,044
2012-13 Boston Celtics NBA $11,566,265
Pierce:
2007-08 Boston Celtics NBA $16,360,094
2008-09 Boston Celtics NBA $18,077,903
2009-10 Boston Celtics NBA $19,795,712
2010-11 Boston Celtics NBA $13,876,321
2011-12 Boston Celtics NBA $15,333,334
2012-13 Boston Celtics NBA $16,790,345
I would never blame anyone for taking as much money as they can but you have to realize that a team can only go over the luxury tax by a reasonable amount before it becomes too prohibitive.