How do I Bevel an Edge within an Edge? Edit Update: 

I want to explain how I got to this point better because I did a poor job explaining, so here it is. I am making a box. Along one edge of the box I pressed "W" then "subdivide" to create a single vertice. After which I had selected the single vertice I created on its own, and then pressed 
"ctrl shift B" to create two vertices, when I had the desired distance I wanted I then switched to edge select, and selected the edge between the two vertices I created then pressed "ctrl B" which results in the image above. I have no doubles, and using "ctrl A" in object mode doesn't fix my issue.
I dont have enough rep points to comment, but, to comment, I did what you wanted, just with CTRL+B, not sure where the problems is with your bevel. Maybe, CTRL+A first ? scale / rotation / location....

OP edit: SpaceKat had recommended resetting my settings. Which I did by reinstalling the latest version of Blender 2.79B, the original version I used was 2.78. Using the newer version of Blender fixed the issue right away no problems at all. I wanted to give SpaceKat proper credit so I edited the post.