The fatigue for lack of rest theory is quite bizarre. The best series for Pierce last season was the Finals and he had much more minutes in his legs than he has this season.
I think the theory is cumulative. In the season before last year's championship run, Pierce played in a total of 47 games, played 1740 minutes, and saw his season end on March 28. If you buy into the "rest / fatigue" theory, Paul should have been well-rested even late into last year's playoffs, because he had such a light season the year before (especially because many of the 47 games Pierce did play in were low impact type games for a team going nowhere).
This season, Pierce is coming off a season (last year) when he played in 106 games (26 of them highly intense playoff games) for a total of 3,864 minutes. Pierce's 990 minutes were more than any other player in last year's playoffs. Along with that, his period of recuperation was shorter, as the Celtics' season didn't end until June 17.
Additionally, this season Pierce was playing high-leverage games all the way up until the beginning of the playoffs, while last year that wasn't the case. Last year, Pierce averaged 32.6 minutes per game in March and 27.4 minutes per game in April; this season, he averaged 39.5 and 34.2, respectively.
I don't think it makes any sense at all to say "Pierce was fresh in last year's playoffs, so he must be in this year's, too". That may be true -- I don't have enough information to say for sure one way or the other -- but if Pierce is a little more tired this year, it certainly makes sense.