I am interested in this topic too... I have a HandiQuilter and the same thing happened to me (also a Janome 1600P). It seems I remember from a previous topic that this Janome model is picky about the kind of thread. I spent quite a while looking at threads online today. The Superior thread site said that problems like this are often mismatched thread size and needle size. I am going to try another brand of thread when I get another top ready to quilt. The last one is sitting on the shelf because I need to remove all the stitching since I have all those bird's nests on the back.
Will be interested to see what everyone has to say.