It might be birds, especially the corvidae ... crows, jays, etc. Another possibility is snails and/or slugs. But remember that rabbits and gophers can easily dig under fences. Also, I don't think there is any fence that is actually proof against mice, voles and rats, which can go over, under, or through them.
"Isn't it enough to see that a garden is beautiful without having to believe that there are fairies at the bottom of it too?" - Douglas Adams