We have 3 cats and new green-cheek conure. One of the cats has successfully lived with birds before, the other two are novices. So far, the experienced cat has marked the cage stand and watched the bird a little, then proceeds to ignore him. One of the others seems puzzled and cautious, but shows no inclination to bother the cage and the third is terrified of the bird! The bird seems unconcerned, at least when in the cage.
For now, anyway, we close the door to the room where the cage is housed whenever we aren't home. It's a pretty secure cage up on a stand, 18"x30"x25" with fine bars...it has a playstand on top that is detachable but sits up there when the bird is inside. I don't think the cats could reach into the cage or knock it over... how much would you worry? What's your cat/bird routine?