Harry Potter: man with hardly any neck

At the beginning of Harry Potter it talks about Mr. Dursley, about a man with hardly any neck

I think I understand the meaning of this phrase: "he has a thin neck." But I can't stop thinking about the expression "hardly any." On what principle does "any" bring a negative connotation here? Are there any other examples of expressions like "something + any"?

The description of Mr. Dursley indicates that his neck is not a clearly defined feature in his appearance. This is not because it is thin but because it is so thick that it is difficult to see where the neck ends and the head begins - to a casual glance his head almost appears to be connected directly to his torso.

Other answers to related questions, as indicated in the comments, address the more general issues mentioned in the question.