I have two answers for this…
First, I’m 38, I’ll let you know when it happens, I still love to play and have fun, go to rock concerts, play video games, and generally slack off from responsibility where practical.
Second, when the crushing weight of real world responsibility starts to take its toll on your body and mind, be it jobs, kids, you name it. I still didn’t feel like an adult in my 20s so much, but once I hit 30 I got married, had a kid, contracted diabetes, and survived a series of layoffs from my job (currently in my 6th month of my latest stint of unemployment). Nothing drives it home better than having a ton of bills, going further into debt every day and having a family who relies on you for financial as well as emotional support. Neither my body nor my brain works quite as well as they once did. I sure feel more like an adult than a kid these days. But yet, I know completely what you mean, I remember not that long ago going through my head that I’m really in charge, I’m really supposed to be the grown up making the grown up decisions, and having that though kind of frighten me.
I guess the real answer is, some day it will hit you, and life will never be the same (not to say it will be worse, just different).