Here is the problem I faced... While editing and moving vertices it appeared that randomly it would select multiple vertices and move them together. Again, it appeared to be very random.
In reality this is what was happening... As I was selecting vertices and moving them I noticed that if I was not absolutely accurate in selecting the vertex that it would not change the pointer symbol and I would move the entire line even though I was snapped to the vertex.
So, if you edit a polygon layer, very slowly move the cursor towards the vertex. What you notice is the cursor snaps to the vertex, but the icon on the pointer is still set to move a line. If you move the cursor a little closer you will notice it changes to a box with arrowheads off every side. Now you can move the vertex by itself.
My Idea.... This functionality is somewhat problematic and frustrating and slows down the editing process. If my cursor snaps to the vertex then I should also get the vertex edit tool to become active. This would make the editing experience more intuitive and logical.