If you miss a train because of a delay, then you can take the next connection at no additional cost. That also means: no additional travel day. The best is then to ask the staff what you need to continue your journey the next day without extra costs. In such a case, they also have to provide accommodation for the night because you would be stranded.

Thank you very much!

DO note that DB trains are often late, very late-even though all old printed guidebooks tell its german efficiency, the last 2-3 yrs it is not. You had better take an earlier train-they run hourly on B->HH. Whereas rian is right in telling the rules, it will be a considerable effort and bring heaps of problems to add an extra day-it usually means you have to pay a normal ticket and hope and pray for refund monthes later.

IF it happens, never do it on your own-lay your hands in those of the DB-service desks which are at any main station. From others I got feedback that IR-passholders will usually in such a case be offered a voucher for hostelbed, no hotelroom, as it fits more in the style! DB has a kind of contract with AO hotels, which are all very close to stations and offer both types.

Actually I had the same prob last sept with an other link to HH, but as I had a continuous pass, I 

´only´ had to rebook all and spend an awful night on local trains and waiting between HH and danske border. End yes-I was indeed offered a bed in a hostel at Bremen.