Stupidity: The super powerful Romulan commandos (no doubt with the help from the higher-ups infiltrated in the Federation) managed to touch up the footage of the fight in the roof so to remove every trace of the Romulan agents present. But Picard has been running, dodging blasts, ducking, and is knocked back by a huge explosion caught on camera and not deleted; we see it in this episode and it is unequivocally not caused by him. What is the point of going through subtly deleting every trace of the aggressors when it is transparent that something real happened and it's not all Picard's imagination? And the police simply take Picard back home from San Francisco to France without questioning him, not caring about investigating the explosion in the heart of the city involving a famous Admiral battling invisible enemies. If the conspiracy can erase anyone from the tape, why not erase Picard too, or the explosion and explain how he ended up K.O.'d differently?
Remembrance - S1-E1
Stupidity: In this episode wily old battle genius Jean-Luc Picard and amazingly smart human-android Dahj act exactly by the definition of stupidity of this website ("something daft, like running upstairs with a killer behind them, instead of out of the front door"), and even surpass it, because the killers are not even chasing them yet. And why not? Because they are in a public area with a ton of people in the middle of what is basically the capital of the world; no band of kidnappers would attack at that point, or at least, it's way more unlikely. But from there, our nearly centenarian hero (steps away from official government buildings and in a world with communicators, teleports etc.) goes up a ramp of stairs leading to a desert rooftop with no exit and no witnesses, exactly where a group of evildoers would attack - and are even able to cover their tracks up exactly because of this choice. (00:31:50)
Suggested correction: Picard is being led by the extremely combat-effective android. She has previously fought off these attackers with ease, and had succeeded again. It was only the exploding rifle that stopped her escaping. Picard and Dahj would rather lure attackers away from a populated area in order to protect bystanders, since the attackers were coming either way.
The slowly-exploding rifle somehow unavoidable for the super-fast android that dies from the barf that a middle-aged caretaker shrugs off would deserve a stupidity entry of its own, but back to the point: if the attackers came their way, they would have never been able to erase their traces by deleting footage, and therefore they would have been the 'stupid' ones. This is pure movie logic and plot convenience, just like the designated victim in a slasher running upstairs rather than screaming bloody murder in the street where they can be helped or dissuade the killer from getting into unfavourable situations.
The End Is the Beginning - S1-E3
Stupidity: The Chateau is stormed by the Zhat Vash commandos. The two trusty housekeepers prepare to fight the aggressors armed to the teeth by grabbing... a bread knife and a wine bottle. All is good, since the bottle is super-effective, knocking down a Romulan wearing a helmet. This absurdity aside, moments later during the fight it turns out that the valets had weapons stashed everywhere in the room, even strapped underneath tables! Why would they start off with such ineffectual weapons instead of reaching for the guns, other than of course for dramatic purpose?