Is it possible to be addicted to antidepressants?

Not exactly: There is no true addiction to antidepressants. However, some are hard to stop due to "discontinuation syndromes"... Side effects that happen when the drug is stopped. There are ways to make this less of a problem. Also, some people are psychologically attached to their antidepressant; they're afraid to ever stop it even if feeling well for years. This is emotional, not physical, addiction.

No: None of the antidepressants are addictive. You do not develop tolerance as you do with addictive substances even though the same dosage may not be effective indefinitely. Many people need to be on antidepressants for the rest of their lives in order to prevent future episodes of depression. There are many different types of antidepressants and it is always a matter of finding the right one.