Bamberg's backpacker hostel is a well-kept affair, with clean dorms, a fully functional kitchen and a quiet, family-friendly atmosphere (staff stress this is not a party hostel). Make sure you let the hostel know when you're arriving, as it's left unstaffed for most of the day. It's located 400m north along Luitpoldstrasse from the Luitpoldbrücke.
Backpackers Bamberg
Bamberg