Well it depends on what level you want to work.
If you want to do everything yourself, you can do it however you like.
If you intend to collaborate with people you know, as a common project, you could split tasks so that each one does what he is best at.
You could create your studio if you’re up to it and set up the tasks the way you like it. The “author” of an animation takes care of storyboarding and editing more than 3D application themselves…
If you want to be hired in a studio, first most studios won’t use blender (although that tends to slowly change), and will have a specific workflow. The director will probably have basic knowledge of all tasks, but won’t be specifically good at them. An Art Director has to be good at all tasks, sorta speak, and be able to lead the team with an artistic vision.
Then you have specialists who achieve specific tasks such as modelling, rigging, animation, texturing, and many more tasks depending on the size of the studio. 3D generalists are employed in smaller studios, in that cas yes, you’ll have to master as many tasks as possible and as well as possible, I really think not knowing organic modelling can be a big obstacle, but it’ll depend on what sort of work you want to do!
Keep in mind that making professional 3D animation can be enormous work, and teamwork as well as a good workflow is the key.
Check out blenderella tutorials, they’re really perfect for learning organic modelling. Clear and concise.