I've never run one but I've played in a couple. A friend once ran a court game for like 10 people, it was pulling somebody aside to scheme behind people's backs. In the end, my character (a prince) and another (my in-game brother) ended up poisoning each other at the banquet since we had come up with the exact same plot.
When I got really obsessed with L5R back in the day, I'd have occasional court sessions. In hindsight, most were kinda boring and existed because I wanted to be a dick and punish the players who ignored social skills. The interesting ones had some sort of other element like trying to protect an NPC from assassination.