You may encounter PJ basses, which are Precision basses with an additional bridge Jazz pickup. They often have thinner necks than traditional P basses, and are marketed as “best of both worlds”.
Well, if that would be true, every pro player would have a PJ and nothing else, which is not the case.
Neck pickup solo: You may think that this will give a traditional P sound, but it usually does not. The chunky P neck is part of the low-middy P sound. You can’t get that sound out of a skinny PJ neck. Plus, the additional components on the bass affect the sound as well.
Jazz pickup solo: Depending on the pickup position, you can usually get a good 60’s or 70’s J bridge sound – typically heard from Jaco. Not the typical choice for support playing, but a good choice for soloing, chords and articulation.
Pickup blend: Beauty of a J bass is that you can blend the pickups in various percentages to get different sounds, the JJ pickups complement each other nicely. However, on a PJ bass, you can’t blend them so nicely – the P pickup will dominate most of the settings and E-A strings will sound different (dull) than D-G strings (ringy). So, forget the wide palette of JJ blend options. In my experience, blending is only useful to add some high-mids to the P pickup.
In summary, PJ basses that I tried lean towards the P and J simultaneously, but can’t really sound like either. I’d rather have a separate P and J. The only PJ sound I loved came out of a Yamaha BB2024X; but it was heavy, noisy and the neck was uncomfortable (for me).
However, I didn’t play all of the PJ’s on earth and preferences are subjective. Victor Wooten plays a PJ, so it should work just fine for many others as well.