I found that, when preparing to baste the quilt sandwich, making sure that both the backing and the top are stretched smooth and fairly taut helps prevent future problems. I put pins every 4" or so, & use the smaller (1") pins, which hold more tightly than larger ones.
Stabilizing well can help. It *seems* that diagonal quilting in each block would be enough, but apparently it wasn't in this case. I've done queen-size quilts with NO stabilizing aside from the basting pins; I just quilted from the center out to the edges & had no problems.
As Maggie suggested, it may be that your quilt wasn't quite square before you put the borders on, and your borders weren't even. This will give you a little "wave" before you even start quilting. Flat is good! Follow Maggie's suggestions in the future, if you haven't done so in the past.
If you do run into a situation like the one you describe, you may need just to adjust the basting pins before proceeding. This isn't against the law, and may help. But, if you have a lot of excess fabric, this won't solve the problem. If you have a busy fabric (or even if you don't), you can often quilt in a pleat or two to take up excess fabric; the pleats won't be obvious in the final quilt if you do them neatly.
Good luck!