This is partially correct. The train operator is not relevant; the tickets are valid with any operator.
However, whether you can use any train depends on the type of your ticket. Only tickets at normal fare allow you to use any train at any time. Saver tickets and Supersaver tickets (e.g. Sparpreis Europa) are tied to the itinerary as printed on your ticket. You must adhere to that itinerary. Travelling with a saver offer, you are only allowed to use another connection in the event where the previous train was late. If you miss the connection, and it was your fault because you went shopping and missed the departure time, you have to buy a new ticket.
To provide you a more detailed answer, please share your itinerary with us.