How do you know it's not adding a day, you aren't printing out anything and you a setting 'today' to the constant Calendar.DAY_OF_MONTH and not to the current day.
Try running this:
Incrementing the date, auto increments the month too!
Only if you use add(..). If you use roll(..) larger fields are not affected.
BTW I would avoid using the Calendar.DATE constant and use Calendar.DAY_OF_MONTH instead. To Calendar they mean exactly the same thing but the Calendar.DAY_OF_MONTH constant name gives a much clearer meaning of what the constant represents.