Can I get a cat?

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.