This phrase often implies irritation or annoyance when someone feels restricted by another's presence or actions. I'd join you, but I don't want to cramp your style with your new friends.
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Emotional Connotation
Can be used humorously or lightly between friends, suggesting they don't want to interfere too much. Go ahead and decorate the room; I won't cramp your style, I promise!
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Cultural Usage
In Western culture, personal freedom is valued, so this phrase may be common in contexts valuing individuality. She’s a free spirit – having a strict boss would just cramp her style.