Well for that to happen you have to consider the term "adult". What does it mean to be an adult? Typically psychologists claim the term means when an individual can live on their own, pay their bills, and take care of themselves without any aid.
With this definition, some people never turn into adults. However, the age this typically occurs is in the 20's, normally 25. So I don't tell myself that as I do not consider myself an adult, even though the government and society do. And they don't really count for much. They both say "happy holidays" (annoying me) and claim pizza as a vegetable... So to my view, I'm not an adult.