When you write #include "foo.h" I would expect the compiler to check the directory of the file doing the including (as opposed to the current directory) first, and if not found there then fall back on a search of the list of paths as in the case of #include <foo.h>.
Unless an absolute path was specified, #include "/foo.h", in which case only the absolute path needs to be checked.
Are there any C compilers where the rules are different?