Why can't I log in to Messages using my preferred Apple ID
Trying to use an Apple ID with an expired password will result in the inability to authenticate an Apple ID for the messages app and will prevent you from logging into other iCloud services. Unfortunately, the iOS device doesn't notify one that the password has expired. Trying to login into iCloud.com using that ID will also fail without any explanation—it just fails.

The iTunes Store seemingly continues to accept the expired password for some period of time after iCloud ceases to do so. (I imagine this is for a limited time only and that the iTunes ID database syncs with the main database regularly.)

If you're having problems you can check the status of your Apple ID and change your password by going to Apple's My Apple ID page and logging in. You will be notified if your password has expired and be given the opportunity to change it.

An AppleID with a time-expiring password is an outlier situation. I assume this is the reason that iOS/iCloud return incorrect/forgotten password errors instead of notifying the user that the password needs to be changed.