I've seen cats hanging around in various parts of the game, but they don't seem to be interact-able. I like Dogmeat but I'm more of a cat person. Can I get a cat for my settlement?
As of the latest DLC, the Wasteland Workshop DLC, you can build a cat cage in any settlement to capture a cat.
Here's a video tutorial explaining how to use your cat cages. Then you can have cats freely roaming around your home!
Sorry to kill your dreams, but as of right now you cannot get a cat "companion" type character in fallout 4. However, when the dev kit comes out in early 2016 I am sure that someone will make a cat mod that can be used on PC and Xbox One. (PS4 mods pending).
There is a cat that's still walking around after I killed everyone in the Covenant and made it my primary settlement. I consider it my pet cat now.
Wasteland Workshop. Make Cat cages and set them on a switch. Go away for some missions and open the traps when you come back. Not companions. Pets. House Cats. Or DeathClaws patrolling settlements.
Well, like I wrote in my earlier (now oudated) answer, you can get a cat on Xbox One and PC due to the addition of the mod kit. Here is a mod I found: https://mods.bethesda.net/#en/workshop/fallout4/mod-detail/1020767