With a June use year, your 2001 points are good for reservations between June 1, 2001 and May 31, 2002. Your 2002 points work for reservations between June 1, 2002 and May 31, 2003.
Since you banked your 2001 points, they are now good for a stay between June 1, 2002 and May 31, 2003. So you do not have any 2001 points available to use for the May 2002 stay (which falls within your 2001 use year). Therefore, the only way to stay in May, 2002 using your own points is to borrow some of the 2002 points.
Remember that banked or borrowed points cannot be returned to their original use year. The points you borrowed for the May, 2002 reservation expire May 31, 2002. If you have to cancel your May reservation, you may end up losing those points.
Best wishes!