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