Where you have a vertex selected inside the red square, the highlighted edges are the ones connected to the selected vertex. The other edges don't appear selected as there is a second vertex that appears to be in the same position as the one you have selected. If you move the selected vertex you will find it moves away from the other one.
The easiest solution is to select all vertices and Remove Doubles. If that doesn't merge anything you can press F6 and increase the merge distance. Don't go too high or you can merge the entire mesh.
Another option is to select the nearby vertices and manually merge them together with AltM
When you get vertices close together you may find it easier to switch to wireframe display to select the vertices.