You know javascript if you can use Array, Number, String, Date and Object effectively. Plus points for Math and RegExp. You should be able to write functions and use variables (in correct scope, i.e. as 'methods' of an object).
I see some comments about knowing closures, extravagant function syntax, blabla. All that is quite irrelevant for this question. That's like saying you are a runner if you can run the 100m dash under 11 seconds.
I say it takes maybe a couple of weeks to become proficient in javascript. After that it takes years and dozens of books and thousands of lines of programming to become an expert, a ninja, etc.
But that wasn't the question.
Oh, and the DOM is not a part of javascript, and neither is jQuery. So I think both are equally irrelevant to the question too.