I feel clutchness is a thing. I know it's dismissed as a sports cliche, but why? Different people have different abilities to handle stress. Some people panic, some people are in the middle, and some people don't care at all. I don't really think anyone RAISES their level of play, but when they stay consistent, while everyone around them folds, it may seem like they raised their level of play.
With Rodgers, early in his career, I don't think he handled stress as well as he does now. Yes, in 2008, Rodgers did have those go-ahead scores, only to have the defense give up another, putting him in a "now or never" situation. I don't think he did as well with those now or never situations, where if we don't score now, we lose. It's natural for a young quarterback replacing a legend. Over time, he's gotten a lot better at this, delivering on these opportunities.
Maybe there's still a residue of this type of thing in the playoffs, unless he goes "unconscious". It seemed this was happening in the second half of the Cowboys game, and most notably against Atlanta in 2010. Otherwise, he's still very good in the playoffs, but it appears to me that he sometimes misses throws that he normally makes easily, and then gets mad at himself or looks skyward. This too is understandable - he's an emotional dude and is smart enough to know what's at stake, and actually care.
Perhaps as his career progresses, he'll get better at blocking out the emotions of the moment and do a better job of getting into Flow, when the playoffs roll around. Again, this is not to say he's not already awesome. Just an improvement theory.
Flow: http://en.wikipedia.org/wiki/Flow_%28psychology%29