Can you please provide name(s) of the persons, birth years and better estimate of when they left Norway? USA censuses of 1900, 1910 and 1920 can help you with that information.
In the 1900 cencus there is a farm called Rime in Buskerud - nowhere else. And in the entire country there's six persons carrying this name, all from the Buskerud/Oslofjord area. I.e. Bergen is not the obvious choice to sail from. So, as Hopkins say - much more information is neccessary.
AND there is absolutely NO persons emmigrating from Bergen 1874-1930 by the name Rime - not unexpected....
There are 14 Reime's sailing from Bergen, residences Voss, Tysnęs and Bergen. There are farms named Reime in Haa Rogaland and Voss Hedmark but like Askeroi and Hopkins said need more info to narrow it down. Carla