I've never run a cable this way, but I would imagine that it depends on the bottom bracket. (You already know that, which is why you're asking!)
Here's a quick look at a bunch of different models of cartridge style bottom brackets from Google Images:
From this quick look, we can see that there is a differential between the internal diameter of the bottom bracket shell (into which you screw the ends of the bottom bracket) and the diameter of the bottom bracket cartridge's "waist". For some bottom brackets, like the third and fourth ones in the second row (Velo Orange and RPM respectively), that diff is pretty small, so there won't be a lot of room between the cartridge and the frame. For the top left and top right ones (Sugino 75 and Shimano UN55) there appears to be more room. Whether it would be enough to run cable housing without putting too much pressure on it, I can't say - again, it would depend on your particular bottom bracket.
If you were able to replace the bottom bracket with the old cup-and-cone style, there'd almost definitely be room, but I can't imagine it'd be easy to find one that fit your particular bike :)
If this is your bike, I found an instruction manual that will help. One of the instructions says to "Work the housing over the bottom bracket shell and up the seat tube until it exits at the top of the seat tube", which indicates that yes, your particular bottom bracket should accommodate this cable run. Note that according to those instructions, you run the cable itself first and then to run the housing over it, which is a bit counterintuitive but it makes sense once you absorb the whole process.