I don't use a HQ but an APQS long arm...but I wonder if your tension is too tight for the cotton? A bad spool of cotton thread is also a great thought by Paper Princess up there... Especially since you moved on to poly which is much stronger and had zero problems. When you set your tension, generally the idea is to set your top tension according to the thread you're using, then set the bobbin case to match. If your top tension is too tight to begin with, you can experience a lot of thread shredding or breakage...even if your tension is even in the quilt itself. I might try loosening up that top tension, then setting the bobbin to match and see if it works for you that way. Also...double check you've cleaned out all the lint in the bobbin area and bobbin case. Cotton thread is VERY linty...I rarely use it just because I hate cleaning up after it! LOL
Even the teensiest of lint balls will throw everything out of whack.