I've noticed that when my internet is down (and I would like to spend time waiting for it to go back up by drawing in Krita), Krita doesn't open. Conversely, when Krita did open, it was a sign that the internet was finally back up. I'm convinced that Krita relies on or otherwise hooks up to the internet.

Any words of wisdom?

I've just disconnected my network cable and krita started and ran with no problems.
What exactly do you mean by 'doesn't open'. Is there no response? A start to respond then fail?
Which Operating System and which version of krita do you have? Is it a standard installation?

The only time Krita uses the internet is to load the rss feed, but it only does that when you've enabled it, and it should not at all affect any other functionality of your computer. It might however be that you have an antivirus or firewall that refuses to open things until it can phone home to ask if the program in question is legit.