Yeah, it works like that.

The negative is 片付かない. You use it when there is too big a mess that it is taking you lots of time to get it tidied up.

「ポケモンカードが多過（おおす）ぎてなかなか片付か ない。」

Remember we are discussing 片付く(intransitive) just because it is what OP asked about. In real life, however, the transitive form 片付ける is more often used.

People 片付ける things.

Things 片付く.　
(The only exception to this is when a parent talks about getting a kid married off to somehwere, which is my example sentence #4.)

Awesome, thanks :)

I think it might sound ridiculous but I am going to give it a try anyway.

下の娘が先に片付いた。残るは上だけだ。
My youngest daughter got married first. My remaining daughter who hasn't got married is the eldest one.

Doesn't sound ridiculous. That is what it means.

Thanks for the help, masaegu.

Is that a very old usage of 片付く? It sounds a bit...old fashioned thinking. Like "I've finally dealt with my daughter" or "I've finally solved the get-rid-of-my-daughter problem" or something. Kind of like how 子供 in all kanji is a bit old fashioned and not politically correct.

My prof. used to comment about 家内 and also the structure of the kanji 安, given its meaning.

No, not at all. The word is still actively used for that meaning, which is why I made sure to include an example sentence like that. I know where you are coming from, though.

Dictionaries will also tell you if it's a very old usage.