A few things to try. (I am assuming this is *not* a top-loading bobbin.)
#1) reset your tensions. This video by Jamie Wallen is for longarms, but his method works well for other machines as well:
http://www.youtube.com/watch?v=Q1mRhcquZTM Just be careful with the bobbin case screw. I loosened mine so much it fell out! (It's a good idea to do this inside a plastic sandwich bag so you don't lose the screw.) I couldn't get my bobbin tension as loose as Jamie's, but I got it as loose as I could before adjusting the top tension.
#2) your needle may be too small. I use a size 16 needle for FMQing on the frame, and a lot of quilters use a size 18 needle. (The needle holes close up when you wash the quilt.) A too-small needle will cause the thread to shred and eventually break.
#3) observe the top thread path. The way the thread feeds off the cone can change as you quilt. I ended up velcroing my thread stand to the machine so it would stop gradually moving off-center as I worked.
#4) how tight is your sandwich? On a frame, you generally want the sandwich quite loose. The rule of thumb is that you should be able to grasp an index finger that is pushed up from below. You may be over-tightening when you roll.
#5) how close is the quilt to the bed of the machine? Another rule-of-thumb is to keep the takeup bar about 1/4" above the bed of the machine.
#6) change to a stronger thread. I have *NO* shredding or breakdage using Glide thread in top and bobbin. Here is where I purchase Glide:
http://www.bobbincentral.com/glide.html
#7) try joining the homequiltingsystems group at
http://www.groups.yahoo.com . There are probably several quilters there with this setup who may be able to give you tips.