It really depends on the breed, age of the bird and size of the run. I think a 6 foot high is more than sufficient for the heavy breeds, although pullets and cockerals can clear it if they want. Once the heavy breeds start to lay and put on a little fat it is too high for them.
Make sure that you have nothing that can give them a boost over, such as a feed can or outdoor roosts. So they can't jump up on the object then up on the top of the fence.
Make sure your fence is straight and well supported. I once had a five foot high run and had to clip wings to keep them in, well I would come home and about half of them would be out running around. I then decided I had to find a hiding spot and watch what they were doing.
There was a spot between two posts that was not taught enough and they would jump/flap their wings and get as high as they could on it and it gave just enough that they simply walked the rest of the way to the top and jumped down to the other side.