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What is the difference between a collapsible chair and a fold-up chair?
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They are the same piece of furniture.
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The kind of metal chair that is used for church halls, classrooms, and other venues is called a folding chair.
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Comments  
They apparently mean the same thing, but neither would be used in everyday speech in the US. The term used is "folding chair."
The term "collapsible chair" is especially bad-sounding - like the chair will suddenly collapse while you're sitting on it. And the word "fold-up" has bad connotations in the US (meaning go broke or fail). So neither of these terms is used in the US.
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 AlpheccaStars's reply was promoted to an answer.