Even though it is considered one of the greatest trilogies of all time, it doesn't mean there aren't any moments of stupidity.
Note: You may put both examples from both the books and Peter Jackson's film adaptations.
The Fellowship of the Ring
During the prologue, at the Battle of Mount Doom, between the Last Elven-Human Alliance and the Forces of Darkness, Sauron shows up to personally turn the tide in his favor, with the power of the One Ring, and a huge mace. He manages to knock several soldiers around like golf balls, and manages to p'wn Isuldur's father. When Isuldur grabs his father's blade, he smashes it with his foot. You'd Think: Sauron would quickly finish Isuldur with his huge killing mace, and kill him instantly. After all, just because you foot stomp and break a blade, doesn't mean it's useless, and can still be used as an improvised weapon. Instead: He reaches his arm out to Isuldur, intending to grab him and burn him, but is too slow about, and Isuldur cuts Sauron's fingers off, including the one with the Ring. It leads to Sauron's body blowing up, and he ends up (after the Ring continued to exist) existing as a flaming eyeball over Bara-Du.
The Cinema Sins narrator even brought this up in the Everything Wrong With The Fellowship of the Ring:
Cinema Sins Narrator: What the f**k is Sauron reaching for? Just kill the guy! You have this huge mace... FOR KILLING.
While Aragorn (aka Strider) is trying to lead the Hobbits (Frodo, Sam, Merry and Pippin) to safety from the Nazgul (aka the Ringwraiths), the group stops at Weathertop to rest for the night. Aragorn goes to have a look around (patrol the farther area), and tells the Hobbits to wait. You'd Expect: The Hobbits to wait patiently and not try to do anything foolish. Instead: Sam, Merry, and Pippin make a fire while Frodo's asleep to cook their evening breakfast, tomatoes, sausage, and nice crispy bacon, which unfortunately gives away their position to the Ringwraiths, even when Frodo tries putting the fire out. The Result: The Ringwraiths attack the Hobbits and Frodo gets impaled by the Witch King's blade, and is nearly killed as a result of it.
After Gandalf gets held captive at the top of Orthanc, Saruman's tower, Gandalf manages to have the moth go get one of the eagles to come and help him escape, which he does later. Saruman is at the top of the tower at the time. You'd Think: Saruman would use his magic to maybe summon a lightning strike or a fireball or something and kill the eagle, along with Gandalf, so that he can't escape. Instead: He just lets him go and makes no attempt to stop him from escaping with the fact that Saruman has allied with Sauron.
When the Fellowship arrives at Amon Hen, Boromir leaves the camp to go get firewood. You'd Expect: Boromir to bring his shield with him when going to gather firewood, in case there's the off chance that the enemy could launch a surprise attack and have protection against long range attacks. Instead: He leaves it behind at camp. And unfortunately, the Uruk Hai launch a surprise attack on the Fellowship and he doesn't even go back for his shield quickly, and when he tries to go save Merry and Pippin, he is left defenseless against Lurtz's bow and arrow attacks, and is killed from the arrows as a result of it.
The Two Towers
The party of Uruk Hai that captured Merry and Pippin camp for the night near Fangorn Forest alongside a group of Orcs who joined up with them not long ago. Members of the party complain about the lack of decent food they've had recently, and a few Orcs consider eating Merry and Pippin. In response, the leader of the Uruks explicitly tells them that the Hobbits are to be taken to Saruman "alive and unspoilt". You'd Think: Everyone in the party would come to accept this fact. Instead: One of the Orcs attacks the Hobbits, claiming that he just wants a nibble. He gets his head lopped off, and the party decide to eat him instead.
After Gandalf has removed the spell on Theoden and he's back to normal, Saruman no longer has any power over Rohan and Grima Wormtongue is thrown out of the court (literally, down the steps). When Theoden goes to kill Grima, Aragorn stops him, claiming "Enough blood has been spilled on [Grima's] account". You'd Expect: Them to imprison Grima to stop him from returning to Saruman with vital information. Instead: They let Grima escape back to Saruman without even trying to stop him. As a result, Grima tells Saruman exactly where to attack and how, subsequently leading to the Battle of Helms Deep, which kills hundreds more redshirt soldiers including children.
The Return of the King
Gandalf goes to Minas Tirith with Pippin, they then meet Denethor, the Steward of Gondor. They tell him that the enemy is going to be on his doorstep. You'd Think: Denethor would listen to Gandalf, who has proven to be wise - if somewhat obtuse at times - and pretty much the one thing that can push the Nazgul back. And even if Gandalf supports Aragorn as King of Gondor, it does not mean that he cannot maintain his position, advising the king on important matters. Instead: He rebuffs Gandalf's quite sensible suggestions and rage about the fact that he'll never let Aragorn take the throne, saying that the throne of Gondor is his, and no others. And in the film, when Pippin lights the beacon that will call Rohan to battle, at Gandalf's prompting, he gets angry with Gandalf.
* Sighs in frustration* The "Faramir being thought dead" fiasco:
Faramir reports that tens of thousands of orcs have taken Osgiliath, which he and just a few tens of soldiers have survived thanks to Gandalf's intervention. You'd Expect: Denethor to listen to his son and realize that Osgiliath is lost for the moment, order the fortification of Minas Tirith and bring reinforcements and call for any allies that may be willing to come. Instead: he rebuffs Faramir's claims, saying that he and his men ran away because of cowardice, and orders him to lead a small army to retake Osgiliath, which leads to disastrous results.
Faramir's attempt to retake Osgiliath ends in failure with the entire unit wiped out, except for Faramir and his horse. Faramir is knocked unconscious from a poisoned arrow, however Denethor gets greatly upset and thinks he's dead. Pippin however notices he's still kicking, but needs medicine, and says, "He needs medicine, my lord." You'd Expect: Denethor, as grief-stricken as he is right now, to listen to what Pippin's saying, which would give him a sign of relief. Instead: Denethor ignores that, yelling "My line has ended!". When he sees Sauron's army at his doorstep, thinking Rohan will not aid him, he yells "ABANDON YOUR POSTS! FLEE! FLEE FOR YOUR LIVES!", which is stupid, so Gandalf knocks him out unconscious with his staff, and tells everyone, "Prepare for battle!".
Since Denethor is knocked out, this leaves Faramir's unconscious body unattended. You'd Expect: Pippin or one of the other soldiers in the area to take advantage of Denethor being knocked unconscious by Gandalf, and take Faramir to the House of Healings, and have him recover, saving his life. ...Or at the very least: Get him medicine, and have him recover. And while they're at it: Since Denethor has gone too far, and has lost his mind, keep him isolated from Faramir's body. Instead (!): Pippin and everybody else completely forget about Faramir, and the fact that Pippin himself said that he needed medicine and everybody, including Pippin (who tries to join the battle), leave his body unattended, and they don't get Denethor away and put him somewhere where he wouldn't have access to Faramir's KO'd body. Unfortunately, this becomes a waste of time for Pippin, who ends up having to go back to the citadel on Gandalf's request. But by then, Denethor regains consciousness, and he still takes Faramir's unconscious body and himself to the Tomb of the Stewards, and none of the other soldiers even question Denethor and his suicidal madness, or ask "What if your son is alive?", or even check for a pulse on Faramir!
It was also brought up by the Cinema Sins narrator in EWW Return of the King:
Cinema Sins Narrator: Let me get this straight, so after Gandalf knocked Denethor unconscious, no one decided to get Faramir any medicine? They didn't just leave him outside in the white tree court, did they?
When Denethor carries Faramir's unconscious body to the Tomb of the Stewards, to kill himself along with Faramir, he tells his members of the citadel guard, "Bring wood and oil." You'd Then Expect (!!): The soldiers of the guard to go, "You're Insane!" and/or "Screw the Rules, I'm Doing What's Right" and not help him kill himself since it's insane, especially since if you end up helping your leader kill himself, you are left leaderless. ...Or at the very least: Check for a pulse on Faramir, confirm that he's alive, and not carry out the order. And go, "Good news, your son is not dead! You don't have to kill yourself, my lord." ...Or even: If neither the above, when Pippin comes in, yelling "HE'S NOT DEAD!", the captain of the guard to at least not ignore Pippin yelling the fact that Faramir isn't dead, and stop to ask Denethor, "Sir, what if he's right (about your son not being dead)?", and then order the guards not carry out the order light the pyre. Instead (!!!): They continue blindly obeying Denethor's orders without question, despite that he has clearly lost his mind, and not EVEN check for a pulse on Faramir! And the guards (including the captain) and everybody else completely ignore Pippin like brainless, deaf morons while he yelling "HE'S NOT DEAD!!" as he's dragged out and thrown out of the room by Denethor, and don't even ask Denethor, "Sir, what if he's right (about Faramir not being dead)?"!
Even worse, in the book, one of the members of the guard (other than Pippin) did go against orders to save Faramir, though even he had to go up against several members of the guard who didn't.
Pippin gets thrown out of the Tomb of the Stewards, with him being fired by Denethor, who tells him "Go now and die in what way seems best for you.", and then closes the doors, but doesn't lock them. You'd Expect: Pippin to open the doors, which aren't locked, and try to convince the captain of the guard, or one of the guards, and put some sense into him, which would maybe cause him to ask Denethor, "Sir, what if he's right (about Faramir not being dead)?", and stop this madness without anyone getting hurt or killed. Instead (!!!): He gives up on trying again, and wastes precious time looking for Gandalf. And it's only by luck does he manage to find Gandalf in time, if he hadn't found Gandalf in time, Pippin would've been too late to save Faramir!
Anti-Villainous example during this whole fiasco: After Denethor closes the doors to the tomb... You'd Think: Denethor would either... A: Lock the doors completly, so that it can't be opened and no one would stop him. B: Barricade the doors, so that if anyone tried to stop him, they'd be wasting their time trying to bang it open, and by the time they open it, it would be too late. Instead: He does neither, and Gandalf barges in, showing up just in time, yelling, "Stop this madness!!" Continuing on: After this, he grabs a torch from one of his guards, and drops the torch into the pyre to kill himself along with Faramir (in the extended edition, before doing so, he says, "You may triumph in the field of battle for a day. But against the power that is risen in the East, there is no victory."). You'd Then Think: Denethor would immediately order his guards to stop Gandalf (either by killing him or just incapacitating him non-lethaly), so that he would not interfere, and then he can get away with (unwittingly) murdering his own son, while killing himself. Instead: He does nothing of that sort, and Gandalf grabs a spear from the nearest guard (who for some reason, doesn't stop him) and knocks Denethor off the pyre, allowing Pippin to push Faramir's body off the pyre and save him from being burned to death. When Denethor threatens to do something terrible to Pippin while yelling, "You will not take my son from me!", he gets knocked into the pyre by Gandalf's horse, and gets set on fire, and runs all the way out and jumps to demise.
In the film version, Theoden and the Rohirrim have just shown up, and proceeded to drive off most of the Orcs attacking Minas Tirith. Then they see Mordor's reinforcements: a huge row of Oliphants/Mumakil (essentially gigantic war elephants) manned by the men of Harad. The Rohirrim consists entirely of mounted soldiers and archers. You'd Expect: The Rohirrim to stay away from the Mumakil and attack their riders from longer range by shooting at them with arrows, or (like what Eomer does during the below "instead" moment) throw spears at their riders, to try and send them out of control. Or at least: The Rohirrim to go inside the ruined city walls where the Mumakil cease to be a tactical advantage. Instead:Theoden has the entire host of Rohan charge the Mumakil, and we're treated to a montage of them getting trampled, sent flying by the Mumakils' tusks, or shot by the Harad archers riding the Mumakil. Ironically, this bout of stupidity doesn't get Theoden killed; it's the Witch King who does him in.
Finally, the eagles. You'd Expect: The Fellowship to use the eagles to fly to someplace near Mordor. That way, they could reach Mt Doom very quickly to destroy the One Ring and bring the enemy down. Instead: The Fellowship (and to the extension, after the break up, just Frodo and Sam) go on foot, which takes days and days, obstacle after obstacle, to get to Mount Doom, which makes the stakes very high. And it's only by pure luck does this plan work.
The Hobbit: An Unexpected Journey
During the battle of Moria, Thorin manages to cut off the hand of Azog, the Orc leader. Azog proceeds to clutch his wrist in pain for several seconds. You'd Expect: Thorin to take this opportunity to finish Azog off for good. Instead: He just stands there, apparently assuming that Azog will later die from the wound. Azog escapes the battle, survives his injury, and comes back with a vengeance when Thorin tries to retake Erebor.