The reason why the Spurs haven't made it to the finals since '07 was that Ginobili, the Q4 closer, their *Paul Pierce*, was injured. And that was a consistent trend, which continued on, postseason, after postseason, since that year.
And now, he's in the declining phase of his career. Thus, the Spurs do not have an all star, who can take over in those key situations. Duncan is no longer the man for that job; his job is pure defense and using that to generate some offense.
In contrast, the Celts have never had a truly injured Paul Pierce during a playoff run. Replace Doc with Pops and we win in 2008, 4-1 against the Lakers, in 2010, 4-2 against the Lakers, and finally in 2012, 4-2 against the Heat in the ECF and 4-1 against OKC in the finals.