Is it possible that a school only has one student enrolled in a particular undergraduate degree program?
Possible but uncommon.

The liberal-arts departments at MIT, for example, generally only have one or two students actually planning to get a degree in that area (usually, of course, as a double degree alongside something in the sciences or engineering.) Friend of a friend was the sole music major at one point. Small department since it isn't the school's primary focus, but I'm told the professors are excellent and of course you get a heck of a lot of attention from them. Of course first you need to get into MIT.