This was my first impulse as well- allow overly skeptical characters to occasionally be too vigilant for their own good, with false alarms and such. However, I was reminded of a discussion I saw about D&D where a DM gave his PC’s with high passive perception scores useless information so that they wouldn’t only notice purely helpful facts. There was an argument about whether this was fair, because it was, in a sense, punishing a character with unique, pointless data just because they had invested in a skill. And I’m torn on that, because on the one hand, most people wouldn’t enjoy it if having a high strength stat meant they would accidentally break things, because they were looking for a benefit for their investment, not unique punishments. (Though I could see a game with a message about the importance of the Golden Mean or something having punishments for going too high and/or too low on stats, I think most players would be unhappy with that sort of arrangement.)
However, it would also suck if every time you noticed an extra detail, you KNEW it would be important, or at the very least helpful, because otherwise why would they throw it in there? It’s a complex problem, I think. I suppose it essentially boils down to how you reward a player without ruining challenge, but with more of an investigative and social spin than the typical combat one you find in game design.
Still, good point, and something I will consider.