The order of the Jedi are not restrictive on what species becomes a member of its knightly ranks. But you must be Force-sensitive, of course, and have the required skillset when it comes to wielding a lightsaber if you want to succeed in joining the peacekeepers of the Star Wars universe.
Cats, it seems, don't meet either of those requirements, being neither one with the mystical energies of the Force or any good with a lightsaber—in fact the don't take much interest in learning anything about this ancient guild.
Just take a look at this bunch of disengaged felines in this video, who upon seeing nothing less than the legendary and wise Jedi Master himself, Yoda (OK it might be just a toy Yoda, but still) don't even attempt to show respect.
What they do is sniff at his feet, cower in their cat beds and, by the end, just seem to want to ignore him in the hope he'll go away and leave them to laze around.
Outrageous. Of course it could all be some sort of ploy, lulling Yoda into a false sense of security before they strike. Because they'll probably secretly working for the Sith Lords. We all cats would tend to the dark side.
Dogs, however, like the one at the end of this video, show much more willingness to becoming Padawan.
What does this all mean? Well i guess you could say that cats are their own masters in whatever they do and don't need a jedi teaching them otherwise, while dogs are just eager to please any human looking life form that bothers to take an interest in them.