You are correct. Usually I see this in CVs. It usually means that the person is proficient in C++ but knows a little C as well.
But C/C++ can be a language. It could be a code written in C++ but looks more like C than C++. Some people e.g. use character arrays instead of string class, avoid STL as much as possible, don't use object-oriented aspects of the language, etc. The end result uses some C++ libraries, compiles on a C++ compiler but resembles C more than anything. I think it would be appropriate to call it C/C++.
I mostly agree with your observation regarding CVs and what seeing this in one tells about the person who wrote it. Although I like to think that most proficient C++ programmers respect the differences between the two languages and prefer to call C++ what it is, and mention C separately if it needs to be mentioned.
I don't think the style of your code should change the name of the language you claim to use. If it doesn't pass through a C compiler, it's not C or "C/anything".
Now it is quite possible to use the subsets of C and C++ to write code that conforms to both language specifications and compiles with a compiler for either language. If some project deliberately does this, I don't have an issue with calling it C/C++ to hilight the fact. This however really is quite rare from what I've seen, even if I can name some projects that do make such code (the Opus audio codec is one example).
C/C++ still isn't a language though, so I would very much prefer to call it C in a case like this.
A different scenario entirely would be a project with parts clearly written in two (or more) different languages, which could happen to be C and C++...
I was looking at an analysis of id Tech's Doom3:BFG engine yesterday, and it's best described as this. The author described it as "C, with classes". Quite interesting, so I went and asked some game devs I know, apparently this is pretty standard!
