It depends on how you define "overdrive", what you define as a "rock mix" and most of all how your dirty tone is implemented.
For say, metal, a clean bass tone (with a bump in the low mids) actually seems to work best. Competing with heavily distorted, downtuned guitars it's generally not a good idea to dirty up the bass or you risk disappearing in the mix altogether.
But yes, in general for most "rock" music, a bit of overdrive can help fill out the sound, especially if there's only one guitar.
But there is a difference between overdrive and distortion or fuzz. In general they are just different degrees of clipping. And one thing to keep in mind is that while a bit of grit or dirt may fill out your sound, as you clip the signal more and more you are also reducing the clarity of each note.
Robert Sledge's tone (fuzz from a Big Muff) sounds great in Ben Folds Five but in a denser mix that clipped, mid scooped tone would render him all but inaudible.
You CAN have a heavily distorted bass tone work in a mix with distorted guitars but as King's X does it really requires some teamwork and "frequency slotting" so that you aren't stepping all over one another sonically.
It's always a cardinal sin of guitarists and bassists to slave over a bedroom tone that doesn't work on the gig. Overdriven bass can sound fantastic in a rock mix but the key is making sure that it fits with the tones the other guys/gals have going on too.