I don't think there's ever a moment when it suddenly "all makes sense." I think it's more of a never-ending experience that you just have to go through. I've taken Japanese for 3 years in college (and even got a minor in it) and I still wouldn't say that it all makes sense.
I've noticed however, that I began to pick up more and more parts of the conversations. As an example, I first noticed that I can pick up the easy greeting phrases, "good morning" or "thank you". Slowly, I began to understand phrases like "Where is the bank?" and "I am going to school".
As my classes continued, I began to pick up more and more parts of the language, till the point where I can understand 60% to 70% of everyday speech.
Never has it all suddenly began to make sense, and I feel that if I were to continue my studies in the langauge, it would be the same process, where I will slowly notice myself capable of understanding more complicated parts of the speech.
Long answer short, I don't think it's as easy to just have the language suddenly make sense entirely. I see it as a continual never ending process.