My ex and I have joint custody of our two children. The divorce decree sates that he claims one, and I claim the other. My question is, both children have been living with me for the past two years, with no help from the father. Do I have a right to claim both of them, even though the divorce decree sates otherwise?

Save yourself the aggravation and time - either get your "ex's" written consent so you both don't claim the same child OR go back to Court and request a modification.

If the court does not accommodate you, you MUST abide by the divorce decree, which is a LEGALLY-binding document.