I think rank & lineage is a big deal. Rank is a big deal because if you are learning from a 1st degree black belt you aren't going to go very far in that system before your going to have to find a new teacher. If your studying under a 7th degree black belt your obviously going to be able to go further in that system. I think lineage is just as inportant as rank too.
So in this scenario, we're assuming that the 1st degree has his/her own school? So if thats the case, and if the school isn't a belt factory and makes the student put in the time and make sure they really know the material, then by the time they reach the BB level, its very possible the 1st degree could have moved up a level.