It's complicated. [emoji14]
Two different games, two different nobles. The problems were purely narrative, not mechanical, and I see in hindsight I could've managed them better. Still, these situations left a sour taste in my mouth for this background.
In the first game, I was hoping to give the players a tour of my homebrew world with a campaign that required much travel. This would have required the noble to leave his lands while they were still under threat. He stated he "wouldn't be playing [his] character right" if he left before the danger passed. The rest of the players jumped on board with his plan to cleanse the local lands, putting a serious crimp in the pacing of the story.
In the second, the campaign was a parable of the American Revolution in a medieval fantasy setting, with the players on Team America. I warned the player the nobility would become very unpopular early in the story. He still wanted to move ahead with his noble background. When the war broke out, he basically sided with the bad guys. Had to make a new character while we discussed OOCly how to handle the old one. He did end up returning to the party, but his initial decision to stick it out with the old world nobility felt like going back on what we agreed to in our session 0.
In both situations, the failure was largely on me for not correctly anticipating how those characters would engage with their background. Prior to the first campaign I mentioned, I never had a noble in the party for all my past DMing, so I call it more a learning experience than a regret. It taught me that nobles require a greater degree of consideration when integrating them into the story/plot than other backgrounds (YMMV). I do regret the second one, because I knew it was inappropriate for the story, but I convinced myself we could make it work. In the end, we did, sort of. Again, he came back to fight for the good guys. It still caused some resentment on my part, and now I'm wary about letting players choose the noble background in the future.
If it happens again, I'll just have a tarrasque awaken under their castle and devour their family, then go right back to sleep so we can move on with the adventuring.