If this is a VGA monitor and you're driving it with Mame and an standard video card, it is likely an issue where the output of the card doesn't match the expected bandwidth of the monitor.
What you're probably seeing is image "fold over". That is typical anytime you have a second image or line that moves in the opposite direction of the rest of the image when you adjust the screen position.
It is possible that the game you're using is only displaying graphics in a small portion of the screen area and therefore only using a small amount of the bandwidth (towards the center). To compensate for a narrow image, you've adjusted the width of the image to be very wide, causing the image to fill the screen better.
The problem with this is that setting the width too wide will actually cause part of the line that is normally hidden under the overscan area to appear in reverse, on top of/under the regular image. That is because if the system is only using a part of the bandwidth, the extreme edges are still there, and sometimes carry garbage signals, and why they normally should be hidden under the overscan so you don't see them.
Try using different resolutions from the video card, adjusting the brightness/contrast, try different games with differnet resolutions/orientations and see if the problem changes or goes away on other games. Then adjust the games that have problems to strech via hardware to a good resolution and adjust down the width on your monitor.
If these are new monitors then it likely is not the monitor. However old monitors with bad capacitors (or other components) in the deflection circuit could also cause similar problems (e.g. Nintendo arcade monitors have collapse problems on the top edge)
Perhaps Ken has more input?