It all depends on what you're going for, really.
I'm all for overdubbing parts. The recording project I'm working now is the opposite however, at least when it comes to drums. For some reason, I'm wanting to keep it to a single drum performance. Not that the music isn't experimental, and you'd think I would want to overdub a bunch of stuff. But I'm going the simple route because it's what feels right for the music.
I've got an atmospheric song where I'm playing Hot Rods (hot rods again!

). I've got one held correctly playing on the ride and the other flipped over mallet style going along toms and doing one-handed rolls on cymbals.
There's no real reason for this, other than vibe and simplicity. I could have overdubbed mallet parts, but I didn't because it felt more natural not to. Experimentation and overdubbing is awesome, and when it calls for it, I'm all over it.
I think that's the thing to ask yourself. What kind of vibe you want?
Also, think about live performance and whether the overdubbed part is so essential to the song, you don't want it absent during live shows.