Things that remind you of a rat are ratty, and so are things that are generally worn and shabby. Your sister's pet ferret might make you jump when it peers out from behind the ratty old sofa with its ratty little eyes.
If your attic is inhabited by a huge, multi-generational family of rats, it's ratty too. Whether this adjective is used to mean "looking like a rat," "full of rats," or "decrepit and dirty," it's negative. The same is generally true for rat itself, which refers to the rodent but also figuratively means "liar" or "traitor." The root of rat and ratty is unknown.