Short answer: No.
So much of a programmer or software engineer or designer's job is independent of a specific language but very dependent on a certain methodology. For example, Object Oriented Programming is very much about where things exist in memory and in runtime, how accessible they are, and how available they are for later use (inheritance and encapsulation). Implementations differ, such as Java having a top-level inherited class Object whether you like it or not and C++ not having any that you don't specify.
Many languages share features (and many diverge). Syntax differs, of course, and it's trickier when you compare entirely different paradigms like imperative versus functional. But String processing is largely the same wherever you go, once you get little questions of syntax, index, and mutability out of the way. Any language with static typing generally has some method of casting and coercion, but how and when you can do it are up to the language designer.
The language is the tool. You're just putting more tools in your belt. But the tool is only as good as the craftsman who wields it.
Now, how many you learn is a personal choice. I would never learn a language for the sake of learning a new one. Any choice to learn a new language is prompted by either a course requirement (had to pick up js, php, and R for a single class last year), an existing project requirement (if 90% of the work is done in perl 6 before you get hired, so is the remaining 10% after) or an application-specific benefit (R over say Java combined with some sugary library for statistical analysis).
That said, you do want to learn what is out there and available. My college taught a course on this subject, Programming Language Concepts. It was a broad overview course, not a rigorous mathematics or a coding related one, covering the basics of what might be considered practical rather than theoretical computer science. It was a great course and I wish it was something offered in the 1st-2nd year rather than the 3rd-4th. The text was Concepts of Programming Languages. Not saying you have to go out and read this book, but that'd certainly broaden your horizons.