Doctor Who, a 1970s example. In Invasion of the Dinosaurs episode 3, Sarah Jane decides to take pictures of a chained, sedated Tyrannosaurus Rex. You'd Expect: Her to keep pictures to a minimum, not use flashes or anything that could annoy the dinosaur, and keep to the small antechamber of the main hanger where the dinosaur is held. Instead: she jams on that flash hard as hell, and even when the dinosaur stirs, she just waltzes in to take close-ups, naturally awakening the dinosaur, which breaks the chains, because The Mole has tampered with them. She's hit by a falling 2x4 while screamingly trying to open the exit door, which has been locked by The Mole, as the dinosaur bashes the anteroom with its tail. Thankfully, Sarah Jane's Character Development moves her far away from these hapless moments as she gets older.
Also, there's the end of "The Deadly Assassin" serial, in which, after the Doctor leaves in his TARDIS, the two Time Lords he has befriended witness the Master closing the door to his own TARDIS, proving that he isn't so dead after all. You'd Expect: them to at least try to apprehend the Master — if not directly, then by informing the proper authorities. Even if they didn't succeed in getting him, at least they made an effort. Instead: not only do they do nothing, they take the time to speculate whether the Doctor will ever encounter the Master again, as the villain escapes right in front of them.
In The Sea Devils, the Royal Navy has rescued both The Doctor and The Master from their watery prison just in time, and have The Master under guard in a hovercraft. You'd Expect: The Royal Navy to keep a massive armed guard on The Master, ready to fill him full of lead whether The Doctor likes it or not. And, The Doctor might warn them about his mastery of disguise. Oh, and you think the crew of the hovercraft might keep an accurate count of their own members. AND, if that weren't enough, that they would always make sure more than one person is guarding a piece of military equipment on the order of a hovercraft. Instead: The Master is apparently left by himself with one sailor to guard him on a small hovercraft. He somehow has time to hypnotize the salor, place a Perfect Latex Disguise on him and palm him off as The Master's own corpse. Then, as The Doctor and the Royal Navy troops are gawking at the reveal, The Master makes off with the hovercraft. This was the serial that the Royal Navy chose, out of all Doctor Who, to endorse and lend the BBC resources for.
The colonists in Power of the Daleks. Daleks have conquered the Earth twice! Now there are three on a derelict ship. You'd Expect: The colonists would read their history and act accordingly. Instead: They accept these supposed servants at face value.
Doctor Who, new series example: Rose in The Empty Child. A rope swings in front of her in WWII London and she holds onto it. Then it starts moving away. You'd Expect: She would let go of it. Instead: She holds onto it as it pulls her of the roof and is left dangling from a great height.
In The Girl in the Fireplace, at the end, the Doctor wants Reinette to come with him. However, the time window into her time forces her to take ¨the slow path¨ to his next visit. You'd Expect: He'd either bring her through the time window to avoid the problem, or he would use the TARDIS to get her. Instead: He tells her she has two minutes. Two minutes later, he goes back through the time window, but it's been several years in her time, and she's dead by the time he shows up.
In The Sound Of Drums, the Doctor, Martha, and Jack find the TARDIS, which has been turned into a Paradox Machine by the Master. The Doctor states that it's too dangerous to do anything with the Machine until it's activated and he can find out what the exact paradox is that the Machine will create. You'd Expect: someone, most probably Jack with his extensive military background, to suggest the following plan of action: Split up. The Doctor and Martha can go and try to take care of the Master before the Machine activates. Meanwhile, Jack goes off on his own to try to find a way to destroy the Machine after it activates in case the Doctor and Martha fail. All Jack needs to do is find a weapon, get back to the TARDIS, lock himself in, and destroy the Machine when it goes critical. He's not taking any risks in either going off on his own or being in the proximity of the Machine when he destroys it, since he can't die. Regardless of which of the two groups succeeds, the paradox fails and the Master is defeated. Instead: all three go off to try to stop the Master. They fail miserably.
In Evolution of the Daleks, two Daleks take a platoon of Dalek-controlled humans to destroy the Doctor. When the time comes to actually do so however, it's revealed that the Dalek's control of the humans is a bit faulty, and the humans promptly turn on their masters. You'd Expect: The Dalek commander (who was remotely monitoring the situation) to immediately realise that the humans are out of their control, and activate the self-termination devices placed in their bodies as a precaution. Once that's done, the Daleks accompanying them can exterminate the Doctor themselves. Instead: The humans are allowed to carry on shooting at the Daleks for a full minute, and succeed in destroying both of them. It's only after their destruction that the Dalek commander decides to terminate the humans.
In The End of Time, Part 2, Rassilon arrives on Earth via a temporal link set up by the Master, which will soon bring the Time Lord homeworld Gallifrey back into existence. The Doctor doesn't want to let this happen, and threatens to shoot Rassilon with a revolver. Rassilon holds off on doing anything, and then the Doctor wheels around and points his gun at the Master, the death of whom would also prevent Rassilon's plan from working. You'd Expect: Rassilon to take full advantage of the Doctor's back being turned, and to blow the Doctor into his component atoms (which we'd seen Rassilon do to a rebellious Time Lady earlier in the episode). Instead: Rassilon just stands around and does nothing, eventually giving the Doctor time to Take a Third Option and disrupt Rassilon's plan without killing either him or the Master.
Rassilon still has the advantage and it's only a matter of time until his plan succeeds as long as he keeps the Master alive and on side. The Master is somewhat miffed that his own plans have blown up in his face, but is nonetheless willing to join Rassilon and the Time Lords in their ascension. You'd Expect: Rassilon to humor the Master and tell him what he wants to hear long enough for him to keep the Doctor from foiling his plan and then dispose of him when convenient. Instead: Rassilon immediately rejects the Master, saying "You're diseased!" and explaining how he could never join them. This infuriates the Master, who attacks Rassilon and provides the Doctor with the distraction he needs.
In Cold Blood, the Doctor and the group he's stuck with have managed to capture a member of the reptilian race, the Silurians, that have abducted several humans. He wants to use the Silurian as a hostage to get a prisoner exchange and needs her unharmed. Among those with him are a woman whose husband and son have been abducted, as well as the woman's father who has been poisoned by a Silurian attack. The Doctor is planning on going underground to negotiate with the Silurians. You'd Expect: The Doctor, having interacted with humans for so long and knowing that they are emotional, would keep the woman and her father with him so that they can't go Mama Bear or Papa Wolf on the Silurian, with the consequences that would entail. Instead: He leaves these two as part of the group guarding the Silurian, and the woman conducts a violent interrogation on her.