Can I sue my college for emotional distress if their administrative negligence and incompetency leads to my not being able to graduate?
Anyone can sue for anything. Sometimes a plaintiff gets sanctioned for frivolous claims. How are we to know what "administrative negligence and incompetency" means, or how it led to you not graduating?
You can sue for anything. However, if your suit says what you said in your post, expect that your case would be dismissed, and that you would likely be ordered to pay court costs and the school's legal fees for something that frivolous.