Is it rude to ask the status of an application?

I don't think it is inappropriate nor rude to send them an email. It's your right to be informed with your result as time is passing.

It depends. What does your acknowledgement say when you had submitted your application?

If you were told something like "you will be informed your result by XX month/year" and you have not received anything in that regard by that period, then you can write an e-mail to inquire the status of your application.

If you were told something like "only successful candidates will be informed by XX month/year" and you have not received anything in that matter, then you can write an e-mail to your contact person to know the status.

In any case, you can write email politely to know about your application status if you think it is already too late. There is no harm to inquiring application status.

If you're waiting 6 months for someone to write you back, perhaps that reflects poorly in their eyes on your skills as a researcher... seriously, though, you have to be more assertive about these things. If you feel it would be rude, try to make it as un-rude as possible: Ask about applications in general, not just your own; say you want to make sure you haven't missed a reply by mistake; ask whether they can estimate when a reply can be expected (rather than asking whether you were accepted); etc.

But don't just wait for it to happen!