Of course each category is independent. I don't think I am the only FT in China who has students with excellent reading and writing skills, but awful listening and speaking ability. IELTS isn't rocket-science, but arguably Chinese-English teachers are more proficient at teaching IELTS than NES are.
The main reason for this is because they themselves have learned English in a pragmatic way, whereas NES just essentially learn from the street. Often, a relatively fluent ESL has a better understanding of grammar than a NES.