There are some videos of cats stuck inside of bird cages, but none of them show how the cat got in there in the first place. (I won’t use a video if I think someone purposely set it up.)
But here’s a young cat who enters the parakeet’s cage to get a drink of water. I get a sense that the parakeet doesn’t quite know what to make of it. Anyhow, this cat’s small enough not to get stuck.