I started in web design in the mid 90's when it was still very new. I now own a (http://www.9thsphere.com) Toronto web design company in Canada and often get asked this same question. Companies will classify these 3 areas in different ways depending on the size of the company and nature of its division of tasks.
.NET is a server side programming language that is used for generating dynamic pages and functionality that allows a website to do some of the more fancy features that interact with a database. Note that .NET is only one type of server side programming language. There are others like PHP. Some people focus entirely on server side programming. This skill may involve updating existing code to creating custom applications. It typically requires a strong mathematical, logical and analytical mind set.
Web design is the art of designing websites. A graphic designer may design logos, brochures and websites. An understanding of a medium is important to creating the right creative message, however, you don't need to know all the technical details of website programming to be a good web designer. Most companies will have a web designer focus on the creative, while a web developer do the technical development. That is convert the creative to code.
SEO or search engine optimization, is primarily about marketing. SEO is also broken down into other tasks like content writing, optimizing code, building links and others. SEO is the process of optimizing a website for the search engines in order to prove to the search engines that your website is an authority on the topic. The search engines have over 400 different factors that determine a website authority - they are continually changing. SEO can often require the skills of many different fields of a website. Even website speed that can be a factor of the hosting server and the code can help. What I'm trying to say here is that SEO requires multiple skill sets and may be more difficult to decide that you want to focus on everything SEO. If you enjoy marketing, you may want to look as specific areas that help in SEO, for instance.
As you can see these 3 areas are very different parts of a website and in my 15+ years of experience in this industry and hiring many people, I have never seen someone who can do it all well. There are many who understand and can do some parts of them, but in order to do one well I would suggest picking what you enjoy the most.
Hope this helps.
BTW if you want to learn more about best practices around web design and SEO bookmark our blog ;) http://www.9thsphere.com/blog (yes. its a shameless plug)