No rimmed case has ever gained much popularity in a modern bolt action.
I'll agree with that, in general, the key being what a modern bolt action is, and when you are looking...
.22LR is a rimmed round, and quite popular in bolt actions.
The classic military rounds, and their rifles, .30-40 Krag, .303 British, and 7.62x54R Russian were quite popular in their day. The Russian is still very popular, primarily due to the low price.
Bolt guns for the .30-30 have been made, the Savage 340 comes to mind. Good gun in .30-30, but a bargin level bolt gun, and never really very popular.
The .225 Win was a rimmed round, but never got popular, mostly because other rimless rounds did the same thing, or better.
.22 Hornet is a rimmed round, and there are several bolt actions available for it today...
Compared to the rest of the cartridge world, rimmed rounds have not been very popular in bolt guns (outside of milsurps), but there have been some, and some have even survived.