Is there a code, I can't find it, that prohibits slipping new wires past existing wires in an existing conduit?

No code violation unless you overfill the conduit. More of a mechanical problem. Good luck!

Hi there myklien,
Welcome to ECN, mate!.
Personally (coming from a guy in New Zealand) I'd avoid drawing cables into a conduit that carries existing cabling, for the reason that if you aren't careful you can burn the insulation off of the existing cables by friction, you can use all the lube you like, but at the end of the day, you still have to be careful.
Bends in the conduit are where the most problems arise.

No, not an NEC one.

We do it quite often but like anything some commonsense must be used. You can not expect to slide a few new conductors in a conduit that is already loaded. Or as Mike said a conduit that is full of bends.

Many times the best way is to choose one of the existing conductors and use it as a pull rope.

If I need two new conductors I will tie three onto an existing conductor and pull that one out to pull the new ones in.

Go slow, again for the reason Mike pointed out, friction could burn through a stationary conductor.

I'm a retired electrician, and have always advised against slipping wires because of the possibility of insulation burn. CalTrans California's Transportation entity has written in there standards wire is not to be slipped at any time.
I have seen it written into so many job specs that I thought it was also in the NEC.
Thank you to all that replied.

Armored cable (with braided wire armor) is great for this- the armor slides easily along other armored cables, and is very resistant to damage.