Rats and certainly mice are easily able to go under an Agribon floating row cover, and would be less exposed to aerial predators. But only plants in open beds are affected. Rodents generally prefer to feed beneath tall grass, or indoors.
Last summer, I witnessed a starling decapitate a fair-sized tomato transplant, and fly away with the greenery. I've seen them carry 18" stalks of grass up to their nests. Several years ago, I dug a bed near a huge maple tree. In the process, I removed snarls of fine tree roots. I shook off the dirt, and tossed them on the nearby grass. Within an hour, starlings had spirited all the root wads away. They've built nests of such materials inside my chimney, inside the engine compartment of my car, and in other truly annoying locations. They're up to no good.
Bob