Continuity always needs to be right, but sometimes there are deliberate errors. Of course you need a good motivation to do one of these, but I think it's great when it gets pulled of right. One of my favourite ones happens in the fifth series of Doctor Who, in the episode "Flesh and Stone".

In the middle of the episode the Doctor and Amy are in a dark forest (actually the oxygen factory of a ship) surrounded by weeping angels. This is one of Amy's first adventures with the Doctor, she's in a place she doesn't know with a guy she barely knows, so she's scared. On top of that, she needs to keep her eyes closed because she's being infected by a weeping angel. Then the Doctor needs to go away for a while, but right before he goes he comes back to her.



He takes her hand and tells her she needs to start trusting him. It's a really nice showing how their relationship is evolving. It's also a huge continuity error. It's not very visible in the shot, but the Doctor is suddenly wearing his jacket with his sleeves rolled up, he's got a different watch and he's acting a lot more emotional toward Amy.