No, they've actually been working on GT5 for quite a while. An early version of the game was shown at E3 2005, although that early version had little in common with the final game except probably a bunch of the old models that now appear as "Standard" and probably a bunch of GT5's many tracks that are of last-gen quality. Either way, some sort of work on the project goes back at least that far.
Besides that, considering that they really only produced a little over 200 new cars for the game and some new tracks (many of them are not new to GT5 and apparently not even updated for the game), three years would still have been more than enough time. As I've already argued, rivals Turn 10 produced more than that in less time (just two years). Remember, none of the cars or tracks in FM3 are lifted straight from FM2, but rather any overlapping content has been completely overhauled for FM3, so all of FM3's content is actually new FM3 content and not just cheaply recycled. Every single car model has ten times the polygons of FM2 car models, for example.
Alright, I quit.
That was my point. PD showed early work that could have been GT5. Gran Turismo HD was supposed to be a 2007 release, but it was scrapped. The development time on what actually became GT5 was more like three years at most.