All right, many things said here are true, many not so much.
1. Construction and woods do affect tone, but you can get a bassy sounding J even from the cheapest Squier on the market.
2. The setup plays a part here, but not so big. If the pickup height is not correct, the worst thing you may get are some dead spots across the neck or a sloppy E string (up to the 5-7 fret ONLY) But bass is still there.
3. A bass is a bass. It'll sound like a bass even thru a 15w practice amp. So this might not be the problem.
4. Saying that "a Jazz is a Jazz" or "get a P" are not too kind words for them basses. Both can sound muddy or thin. EQ might help a little.
5. Electronics are what you go for to fix this problem. And I don't mean messing with active things, passive will do it. Adding a set of heavy gauge strings (.50 or .55 up) will help too, but there's no need to change the pickups. Believe me, if you WIRE THE PICKUPS IN SERIES you'll get what you want. A Jazz wired in series delivers a FAT tone close close close, real close to Precision. Both pickups will be always on and only 1 vol pot will work, but they're gonna act like a humbucker (this will only happen if the pickups are rw/rp with opposite magnetic directions -- note that this thinness may be a polarity problem: take to the luthier and ask him to check if both PUs polarities are correct; this problem will cause a SEVERE THINNESS on Jazzes). You can add a DPDT On-On switch and make it switchable from series to paralell and there are some tutorials on doing it here in talkbass. You can DIY if you can solder.
Don't waste money if you don't want to... I mean, if you really want to get an active preamp and new pickups is up to you, but try the series wiring first, if you don't like there's always the new pickup option!
