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Sorry, Maybe the pic wasn't clear enough. As you can see, all the faces are quads and there are no extra vertices in the middle of any lines. Except that the edge loop in the second pic doesn't loop at all, because the lowest round of faces are extrudes from the upper lines....
I don't think this is your graphics card. It's more likely to be an extrusion error - have you checked for lamina faces?
I say this because your model doesn't look like you've extruded anything - it's possible that you hit the "Extrude" button and then changed your mind. This will leave the extruded faces on top of the originals, and leave four additional faces around the edges that have zero area. The faces will share all four corner vertices, but will still be separate and might impede the "Insert Edge Loop" tool.
To fix this you might be able to select your entire model and merge all the vertices; you might be able to fix it by using the "Mesh-Cleanup" tool, or you might be able to individually select the offending faces and delete them.
EDIT - I just re-read your post and see that the lower set of faces was extruded downward from the edges above them, which would make my assumptions incorrect. When you extrude, make sure the "Keep Faces Together" option is checked. It's possible that the bottom vertices are actually two vertices sharing the same space, in which case the "merge vertices" option should work. You can avoid this by making sure the "Keep Faces Together" option is in fact checked.
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