Is it possible to be addicted to diet coke?

Yes: Addiction is the state of being enslaved to a habit or practice or to something that is psychologically or physically habit-forming, as narcotics, to such an extent that its cessation causes severe trauma.

Yes: One can have psychological addiction to it. If someone is taking in large amounts of caffeinated sodas or other sources of caffeine and suddenly stop - this can make them ill.