So I just discovered I've been gaming on this television for a few years without actually seeing the whole picture. It only became obvious though with Fallout 4, because when I would go into build mode, certain parts of the HUD would get cut off. No other game I guess had HUD elements that close to the edge, so I never noticed it.
It all boils down to a setting in your television called "overscan", which apparently zooms into the image a fixed amount to cover up certain things in television broadcasts (old signals would have information printed in the screen boundaries, and in the early days of HD you would actually see mic stands and things in the edge of the screen sometimes). I knew it had to be something to do with my television though because the one in the living room displayed it just fine. By disabling this "overscan" feature (mine has apparently been enabled since we bought the TV), I was able to fix my problem.
Here's a few screenshots.
This one was taken with my PS4's share button, and shows what I should have been seeing.
https://dl.dropboxusercontent.com/u/6017319/Images/Gaming/fallout4-fromps4.jpg
These two were taken with my cell phone camera and show what I was actually seeing.
https://dl.dropboxusercontent.com/u/6017319/Images/Gaming/fallout4-overscan.jpg
https://dl.dropboxusercontent.com/u/6017319/Images/Gaming/fallout4-overscan-zoomed.jpg