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This can be invoked by the player through the use of purely aesthetic outfits and cartel unlocks. Buy some truly silly or out-of-place looking outfits on the Cartel... and watch the seriousness of the scenes get ruined because your character is wearing absolutely ridiculous armor.
For a laugh, use the character creator to make an elderly-looking character and see how many times people describe them as "young" or call them "kid".
Some dialogue options are tied to class rather than race. The player can have some fun with legacy or cartel unlocks to have a Cathar player ask "What's a Cathar?" on Ord Mantell, go through the Jedi Knight or Consular storylines as a Sith Pureblood, have a Sith Pureblood Bounty Hunter say "Well good thing I have no Sith DNA", go through Mako's storyline while looking almost exactly like Mako, or have an obvious Cyborg rescue someone from "Cyborg freaks". And that's not even touching all the (often ridiculous) hairstyles you can put on your characters.
If the player doesn't romance Vette, the ending to Jaesa Willsaam's light-sided companion story can end with her accepting the "logic" of having the player's child.
The Sith Warrior's "Force Scream" attack, damaging the enemy by expelling force energy from the mouth. The problem is that it sounds way too deep to qualify as a "scream" - at best, it sounds like a roar, but at worst, it both sounds and looks like a belch.
The reused body language animations all the characters use tends to result in this, particularly how many use the same hand-waving motions while explaining things, or while not explaining anything, regardless of their personality or how natural the motions are for a given conversation. This is particularly egregious with the "turn head to the side before turning it back with a slight shake" animation, which tends to pop up at least once per conversation.
The "Smiling" animation can look between Uncanny Valley and this since characters all have the same smile regardless of their model.
Lord Vanithrast's voice acting in the Maelstrom Prison flashpoint is unintentionally hilarious.
By default, Togruta characters have their headgear disabled... but they will still have a filter applied to their voice unless the helmet is hidden.
The beginning of Chapter VII and IX of Knights of the Eternal Throne begin with the player character waking up after nightmares. Unfortunately, your character is always in their equipment. And will even have their helmets or weapons on them, making one wonder just how they can sleep wearing heavy armor and/or with a rifle strapped to their back.
Playing a Chiss Jedi will result in an inadvertently hilarious example of racial insensitivity◊ when meeting one of the mission contacts on Hoth. The Chiss race is unavailable for Republic classes until you get a Chiss character to level 50 (or pay to unlock the race), so it's a scene that you'll likely set up deliberately, rather than something you encounter by accident.
While Darth Baras is meant to be a serious antagonistic figure for the Warrior, much of it is undermined by the fact that he has Body Type 4. As such it's a bit hard to take him seriously.