As cats are naturally curious animals, we use this expression to suggest to people that excessive curiosity is not necessarily a good thing, especially where it is not their business.
If you cut it fine, you only just manage to do something- at the very last moment.
'Cut things fine' is the same. 'Cut it a bit fine' is a common variation.
If you cut off your nose to spite your face, you do something rash or silly that ends up making things worse for you, often because you are angry or upset.
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