MIE covered a lot of the points pretty well already, but I what I personally dislike about Kimberwickes is that once you make the chain or no chain, and which slot the rein goes on decisions, the bit has no flexibility. You're always engaging the leverage action whether you want to or not, so there's no way to reward the horse for giving to the bit. It's very common to see horses ridden in a Kimberwicke to overflex and go behind the bit for exactly that reason.
Riding in a leverage bit with two reins, such as a pelham or a 3 ring, gives you the flexibility to engage the leverage effect only when you need it, and to ride entirely off of the snaffle when the horse is responding well. So my recommendation would be if you think you need a Kimberwicke, ride in a pelham or a 3 ring or any other leverage bit the can be ridden with both a snaffle and a curb rein; you have a much better chance of teaching the horse how to respond to your aids correctly. The Kimberwicke has a much better chance of teaching the horse to evade the bit entirely.
If your horse's issue is the shape of his palate or the fit of the snaffle, the recommendation for a Myler is a good one as well.
ETA: There's a good reason why they're not a legal bit in dressage, and highly discouraged for hunters. (I don't think they're absolutely prohibited, but I doubt you get pinned using one anywhere but a local show.) I believe they're only an accepted bit in AQHA shows. Someone please correct or add to that if you have more current info than mine.
Last edited by maura; 01-29-2011 at 02:48 PM.