The metal pipe under the bathroom sink broke off in the wall. I cut the sheetrock away and there is a cast iron pipe. The threads of the old pipe are really in there. It is a really old building so in tight.

Presuming you are speaking of the drain you can use a SawZall (or hacksaw if you've got a good forearm :) ) to cut out the part with the threads stuck in it then use a NoHub in it's place.

Here's what we call NoHub (made by Fernco):
http://admin.ibeco.se/images_upload/...or-200x181.gif

Am I right in that the threads are broken off in the T of the drain/vent?

The threads are in a cast iron threaded sanitary tee. You're going to be in for a real job if you cut the stack to replace the tee. To remnove broken threads without having to replace the tee take a small sharp chisel and work it under the broken threads, (don't worry if you nick some of the female threads getting under the broken ones). You can now begin to pry and peel the old threads out. Good luck, Tom