"Snatching" can have numerous reasons, but - most likely- will be related to discomfort of some kind.
It may be the bit itself, the bridle fit, the rider's hands or some other pain issue elsewhere in the body.
"Naughtyness" usually has its origine in discomfort - but has not been addressed by us because we misunderstood the behavior when first displayed.
Have a read here, you might find some interesting clues:
http://www.bitlessbridle.com/about/articles.phtml
Good luck