Did you measure your quilt through the middle before each border, then cut each border to that measurement? I do think the skinny border stretched, but I don't think it would have mattered if you had measured through the middle after adding that border, then cut your last border to that measurement and eased it to fit. The problem in the photo is most often caused by just sewing border fabric on without pre-measuring and pre-cutting to fit.
Edit: Heavily starching the fabric before cutting would have helped the skinny border stay unstretched while you sewed.
Oops! Sorry for the double post.