I have had one situaion of a dropped line - and will never again drive withotu my reins attached to each other.
I was working with a troubled teen trying to get her into horses any way I could. I had taken her out driving down the road a couple of times, she was 14, and as my horse was being an angle, I gave into the pleas to let her drive. I handed her the reins and within three minutes she has stepped him up to a trot, fine, and them promptly dropped one rein!
Her first instinct was to pull on the rein she had - and promptly steered one wheel into the ditch - which really confused the horse. I could see him turn and look at us like we were NUTS for wanting one wheel in the ditch.
Luckily, I just calmly said "WHOA" and he came to a nice stop within five feet and waited patiently while the kid jumped out and got the lost rein. I tied them together (that night fixing the broken buckle) and off we went with a bit of a rough start to get out of the ditch, but otherwise no worries.
I know that without that "WHOA", there could have been a serious wreck and wont drive a horse that doesn't have one!