Was on a Swiss train that did exactly this yesterday. To be fair, the delay originated in Italy (20 minutes late)and there was a minimal impact for people wanting to get to or leave from that city (Bern) as passenger on the affected train could get off one stop early and then catch another train to finish their leg and passengers trying to board the train in Bern just had to catch another train to the subsequent stop (Olten) to get on the late train there.

Finally, this was the first time in over 20 years of light rail use in Switzerland that I experienced this.