I think that this Apple ID/password mess is just too much bother. I don't know how many hours I've spent researching the problem and trying some of the solutions.
Like many of you, I've wound up with multiple Apple IDs; let's call them A, B, and C. If I remember correctly, I got into this because I forgot a password. At the time, I remember that the only solution seemed to be to create another Apple ID. So now I can get into iTunes with ID A, but almost everything on my iPhone, iPod, and iCloud are for ID B. I know that a month ago ID B had a password in a short list.
With the upgrade to iOS 6.1 and maybe before, none of the recent passwords for ID B are acceptable. If I ask to change the password for ID B, it will not accept my birthdate as the first security question and it will not always send me email.
Even though ID B is in the list of email addresses for ID A, there is no way I can use the ID A change page to change the password for ID B.
I think this whole situation violates the Human Interface Principles for iOS of consistency and user control. See http://developer.apple.com/library/ios/#documentation/UserExperience/Conceptual/ MobileHIG/Principles/Principles.html#//apple_ref/doc/uid/TP40006556-CH5-SW1.
I've resigned myself to living without access to iCloud, with consistently clicking "ignore" or whatever when iCal and others ask for my ID B password, and a host of other annoyances. I've already sent feedback to Apple. I suggest you do the same at http://www.apple.com/feedback/, selecting iPad, iPhone, or iPod. Maybe if a few hundred of us do this, Apple will come up with a better solution than the varied advice here that works in some situations and not others.