Your Kenmore should be able to handle it...some things to check on:
- Is your thread old? if so it will break easily.
- Do you have the right needle? A topstitch needle is always a good choice because it has a larger eye and can accommodate threads better than universals and some others.
- Have you checked that you have threaded the machine correctly? When in doubt, rethread ...both top and bottom.
- If you're doing free motion quilting are you moving your quilt sandwich faster than the machine can push the thread thru? This can create extra tension that causes the thread to break.
There are many other things to check, and I'm sure others will chime in with their ideas.