When I am hooked up on level ground, I have around 8-9" between the bed rails and the bottom of the trailer. I don't go off road, but I've been in and out of some pretty hairy places, and thankfully never creased anything.
There are a couple of questions I must ask- What brand and model hitch are you using? Hitches vary, but most will have some vertical adjustment to them, and some feature coupler bodies that pivot side to side to you don't start torquing things up too bad.
Second, does your trailer's pin box have vertical adjustment left in it? On one of my older trailers, I had to flip the axles so the springs were on top and adjust the pin box down to get the truck under it. A lot of work, but I ended up with bags of room between the truck and trailer.
Just go slow into your driveway and watch closely to see if you are going to hit. Sometimes, only a small adjustment of the hitch or trailer pin box will make a big difference. Also, trying another approach (literally) to your driveway may also solve the problem. the straighter the rig is, the greater the angle you can pivot up and down. In some cases, I have made almost a 3 point turn in a road to get my rig straightened up before going over a steep breakover so that I don't get everything twisted up. It is easier for me to get into my driveway from one direction, but not the other.