Easily 16 hours. We've done the drive for the past 4 summers. We leave home about 3 am and traveled as far as we could. First trip, we made it to Asheville, NC before calling it a day. Next trip, we hit bad weather and only made it to Seaverville, TN. It was raining cats & dogs and although we were disappointed to stop early in the afternoon, we felt it safer that way. I think the next year we made it to about the same spot, can't recall exactly. Last summer, we drove all the way to Columbia, SC. Made a very easy trip the next day.
We found when the kids would start moaning and groaning about wanted to stop, putting on a LONG movie really helped to pass the time for them. I can't imagine not having a DVD player for them in the car.
I just bought airline tickets for July this past week. We took a trip to Gettysburg last month and although it wasn't as long as a drive as to Hilton Head, I think we've decided to spoil ourselves and fly from now on. I booked us on Independence Air. I've read varying things so I hope it turns out to be a positive experience.
It's not really a bad drive. Indiana is the biggest state to travel through. And, we've learned to come to a compromise in our car as far as traveling through the mountains. DH drives, I look at the floor and keep my mouth shut.

I was convinced we were going to drive off the road no matter how slow DH drove.