Regional trains in Germany do not have reservations, so any time you buy a ticket, full fare or a Länder-Ticket, you do stand a chance of the train filling up, but in my 20 years of experience traveling by train in Germany, that is a very rare occasion.
Regional tickets and Länder-Tickets are open tickets, so if a particular train is full, you can catch the next one.
The one most notable instance I can remember of a train being full was an evening rush hour train coming out of Hanover. It was SRO coming out of Hanover, but as we got farther out of town, people got off, and there were seats available.
On another occasion, a train from Salzburg into Munich in 2012, when Bayern-Muenchen was playing in Munich for the European championship, I got on in Rosenheim and almost all of the seats were taken. I don't know if the train filled up or not. As we got closer to Munich, passengers had an S-Bahn option.
As Russ says, train station lockers are about a save as you can get, but not all stations have lockers any more.