In Copenhagen you could reasonably plan to go from the central station to, say, Høje Taastrup, but fall asleep and end up in Aalborg 5 hours and 400km away.

Høje Taastrup is a large suburb at the edge of Copenhagen, and the end of an S-train line, but anyone who lives there will know an intercity train is faster. Trains to Aalborg stopping at Høje Taastrup leave at 00:50 and at 02:50 tonight. The other suitable intercity and regional trains overnight are to nearer places, 150km or so.