The only problem I've come across with the Classic Player so far are the pickups... it sounds more like a Strat than a Jazzmaster, and apparently that's down to the pickups on the C.P. model being pretty much Strat pickups... like it says in the Guitarist review. Thing is, the Japanese Fender Jazzmasters have similar Strat type pickups, so to get a Jazzmaster which sounds like a Jazzmaster you either have to get the Am. Vintage one, or replace the pickups in the CP or the Japanese guitar.
Obviously if you don't like the sound of a traditional Jazzmaster, this wouldn't be an issue at all!