Rhetorical questions here. Is it even possible to NOT learn C if one started off attempting to learn just C++, as C is indeed a subset of C++?
A retorical answer then. 
Yes, it's quite possible. I've never learned C and began with Java, later moving on to C++ (through the C++ Annotations link I've given above).
I know about some C-like constructs where they share C++ syntax and functions, but I for example have never learned to work with malloc and friends, C casts, atoi and suchlike, and would have problems with function overloading etc.
I'm sure I would have to delve in the C books to write the same programs in C as I could in C++. Just because C is a subset of C++, does not mean you need to know everything C has to offer, specifically those parts of the language replaced by C++ alternatives.
So it's quite possible to just learn C++ and ignore the depricated parts of the C language, meaning that while you're fluent in C++, you couldn't write the same programs in C.