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I agree! delivery can definitely elevate the status of a pun—but I guess, for me at least, even a highly elevated pun is clever, enjoyable, but rarely funny!
Some puns are funny but the ones about unemployment don’t work >

This pun is so meta—I’ve never seen a pun about puns 😂😂 >

It really depends on the delivery. Just pointing out a pun is lame, but when you don't expect them from someone they can be pretty funny.