Does anyone else actually realise that students start Hogwarts when they are 11? They do not start when they are 13 or 10 they are 11. If you turn 11 on September 2nd you have to wait a whole year before you can start I'm afraid, every single student starts school when they are 11 eleven not 15! And also I don't think it would be very likely that Dumbledore or any other headteacher would let a student who had been home schooled just randomly join in their sixth or seventh year. The only time that would happen is when Voldemort had control of the school because he made it compulsory for all school age children to go to Hogwarts at that point. But that is the only time it would happen!