There's always a special human being out there so I wouldn't say it's impossible but I'd say that it's rare and unlikely to happen. As other people mentioned, we have different brains. 

Grab an english speaking 5-year-old child and a 25-year old adult and throw them in Russia, go back in 10 years and check the results! 

The adult might speak perfectly, and actually way better than the 15-year old boy, but the boy won't have an accent and will probably have a natural native speaker's mind with natural speech and usage of words while the adult will always have " that thing " that keeps them from sounding native! 

I agree with Alan. One huge benefit that children have is that adults intentionally mentor them in their language acquisition, speaking slowly and repeating things often and simplifying their speech to make it easier to understand. They also offer lots of positive reinforcement and encouragement adn typically exhibit more patience than is granted to adults who are struggling with the language. A store clerk might find a child's errors endearing and will make a bigger effort to understand, but isn't likely to be so patient with an adult who isn't proficient. An adult learner also typically already has a native language that would influence acquisition of a second language.

In the first place, it takes a child a very long time to do that.

In the second place, there is pretty good evidence that childrens' brains are different from adults, and there there are critical periods in childhood when the brain is ready to receive a language in a way that is different from the brain of an dult.

In the third place, while I'm sure it is <em>possible</em>, I know many people who emigrated to the United States and have lived in the United States for decades, speaking nothing but English eight hours a day at work, and are functional in English, fluent in English, read novels in English for pleasure, are able to express complicated concepts and to negotiate in English--but never lose their accent and never stop making grammatical errors and non-idiomatic work choices. So, even decades of immersion and the motivation of needing to use the language are not sufficient to learn a language fully. 

I think it's very unlikely that an adult will find him- or herself in a child's situation, in nearly constant contact with caretaking adults with effectively infinite patience for a stretch of four years or so. I also think it's a rare adult who can suspend everything he or she has ever learned in order to return to that beginner's mind. And as Denis says, that would be pretty much a waste of time. Remember also that the brain changes with age. Unused neurons die, and the ones that are used to firing together continue to do so.

To learn a language like a child means to start from total scratch, trying to figure out on your own how sounds form phonetic units which form words, which in turn form sentences and so on. I believe it's possible, but doesn't make a lot of sense. As I understand it, the purpose of formal teaching is to save time and help to avoid reinventing the wheel.