I too agree about cotton thread being a magnet for loose dye particles. I had a hand quilted quilt that I had made my markings with the blue chalk that comes in Chaco liners. The blue washed out of my fabric just fine but had imbedded itself into my yellow cotton quilting thread and turned it a subtle green color.
Do you have any of the thread left? You could always test it by cutting off a good size hunk of it (around a yard or so) and submerging it in a white cup with water to see if you get any bleed. Most likely it is/was the backing fabric.