This happened at Melbourne Uni this year with a slight twist - the student support team emailed all-students@unimelb.edu.au (or some variation) letting them know about the support services the uni had on offer for students in distress.
Some poor soul (or a skilled troll) replied to the list with an email intended for the support team itself requesting help with coping with the stress of exams and uni life, starting a "please remove me from this list" conversation that went on for a day or two before they closed the list.
Honestly I've never felt so bad for someone in my whole life, although some of the replies were hilarious.