Why Does This Happen? I just want both sides to lift up like the one on the left so that I can make a cool paper airplane folding animation. I tried with a plane and got the same issue. I even tried using two loop cuts in the middle and putting them right next to each other. 

Update: 
It may be hard to see but the little dots that indicate the middle of a face are in a bunch of weird spots.
You have overlapping faces in your model. You need to delete the ones you don't want.

If you use the dots in face select mode as a guide, it should be hard to find and delete them.

It looks like you may also have duplicated vertices, so I would suggest also using the remove doubles tool on your model to get rid of them.