This post is based on the article "Developing your Sixth Sense" by Eddy Meechan in TGO magazine (March, 2010). In it, Eddy highlights the importance of proprioception - the sense of body position - to hill walking. This can help prevent stumbles and slips on gnarly ground and is something that can be practiced by exercising standing on one leg... with your hands in your pockets... with your eyes shut. Or you could use a wobble board or get into slack lining.
Another point that Eddy makes is that thin soled foot wear can help you feel the ground and contribute to balance. This runs counter to the traditional view that big boots are safest in the outdoors and adds another perspective on the advantages of thin soled fell shoes. (Please see this post for other arguments.)
Finally, it's also worth bearing in mind that you can lose your sense of balance if you don't use it. And it's worth considering its deterioration with altitude: you may want to factor in extra time and protection for those airy alpine ridges, if you aren't acclimatised, you're ill or hung over.