I wont use a tie-down or bonnet unless I have too - sadly most of my horses run a better pattern with one than without - but can all run without. ANYWAY... I'd use a bonnet on a horse that pops up out of his turn instead of driving forward out of it. If you watch Duke run barrels he's very stiff/straight when leaving and I think (just from observation) that without it he would loose a lot of drive and forward - without it he'd shoot more up than out. If that makes sense? I believe Jane Melby may have used one on Beauty at one point as well. Sometimes people use them as crutches/to hide holes. But with Duke I don't think it has to do with lack of training, some horses just run a different pattern.
I think Lance Graves talks about this in one of his DVD's (1st one I believe)- don't quote me on it though.
Just like with all things, use it properly and honestly and you wont have a problem.