Not sure if this applies to you, but the scroll bars will vanish if everything is in that window dimension. If you are in Icon or Button view, all the other icons could be over to the right off the window. The bottom scroll bar would allow you to reach them. Using the arrange commands would bring everything back inside the horizontal window width, and overflow would be downwards. The vertical (right) scroll bar would reappear as needed.
If you are in List view (implied by your post), then something is actually wrong. I would suggest running TechTool Pro to check the integrity of your system files. Or just replace the system files from backups.
If this problem happens on multiple systems, then something corrupt has gotten around. It would pay to track down the culprit on one system and try to figure out how it got on multiple systems. Could be some custom extension that a student installed on several Macs. Does booting with Extensions Off eliminate the problem?