I used
this style of Craftsman extractor. They, so far, have succeeded twice, and failed once at extraction. Every other type of extractor I've tried has failed. I believe the extraction failures were never due to the extractor used, and were more due to the rust level or tightness of the bolt. However, I don't mess with success anymore and try to use these things when ever I can. Part of their greatness is that they use the power of a drill to do the work for you.
If you're lucky like me the stud just broke off, and wasn't over tightened or anything. In this case, it should be very easy to extract even with something
like this. I know that once I extracted my stud, I looked in the hole and it looked like another piece was stuck in there. This was not the case, and you may be seeing the same thing I was. Check to make sure there's really something in there by removing another stud and sinking a long object down the hole and making note of how far it will go in, then check the suspect hole for comparable depth.