If it was an LCD I would say you have a bad pannel.  The next time that happens, unplug the monitor's signal cable to see if you still see the line over the monitor's self test screen.

I am having the same issue with about 20 of the 100 purchased, and all are setup the same. 

I have the same problem,and Im already on the replacement.These monitors are obviosly defective and Dell has to do something about this.Theres plenty of these complaints and its ridiculas..

Gamitt,

 

     This is now a known problem by Dell now,   Try calling them again..   Hope this helps