Intermittent access issues are something I've gotten used to after years of mobile me and now iCloud. They aren't very frequent but they do happen.
The LAST thing I would do is start changing account settings merely because I got some popup telling me I can't log in. If you can log in on the web site but mail.app is having trouble and you are certain the password you gave Mail.app EVER worked, something is wrong with iCloud. Go find something else to do and resist the temptation to fix something on your Mac's end that isn't broken.
Apple loves to talk to customers on the phone. They are waiting and eager for you to call them about any iCloud outages you experience. Just don't let them talk you into deleting things, re setting up accounts or anything else you don't have time for or don't fully understand. When they start rattling off things for you to try, ask them questions before you do anything because they might assume you have an hour free to debug things if you don't set clear limits first.