Bargellos are tricky to keep the beginning and ending stitches connected after the re-cutting. I use a smaller stitch length if I am going to re-cut units and I handle them carefully. Is the thread coming undone in the other one a Bargello too or just regular piecing? If it's regular piecing, I would try backstitching to begin and end your seams. I always do that if the blocks or sash is on the outside of my quilt. If it is the thread that is breaking rather than coming undone I would switch threads.
Congratulations!I see the mystery has been solved so just ignore my advice.