I have to wonder how much weight it would take to snap a typical bike shifting cable? We have to be talking at least several hundred pounds.

So OP, if you are breaking cables that ought to outlast the rest of the bike (assuming they don't get rusty) something is clearly wrong in your setup. Cable routing being the most likely culprit. I would overhaul your cable and housing and try to figure out where the cable is binding or whatever it's doing.

It's not normal to go through shift cables so fast, and whatever the problem is, it's probably compromising your shifting performance too, so definitely look into it. I would go as far as to replace a brifter if that was the problem, this is serious ****.