The answers to your question will vary from one circumstance to the next. And, not all network administrators followed the same path to their position, In my case, for example, I was the person who took the "computer bull" by the horns, so to speak,and brought our company into the computer age about fifteen years ago. I now have the role of network/computer administrator on top of another role I have with the company. I think there are many like me, those who "grew" into the position.
I don't have the credentials you have, nor do I have as wide a range of experience as many of my peers. What I do have, however, is the ability to apply Information Technology / Computer Technology to the company's business plan to make it more productiveand profitable. And I believe that’s the key to success, and the best way to get your foot in any door. If you can show that you can improve a company’s product, process or profit, you’ll have more than one offer from which to choose.
Look at this article titled, “Career Planner: Network administrator”.
http://home.techies.com/Common/Content/2001/CareerPlanner/netadmin.html
It’s a very general - yet very informative – article that discusses jobdescription, useful background experience, typical projects, necessary training, salary, and much more. There are also further links that may prove quite helpful.
From what you’ve described about yourself and your experience, I would have to think that you could easily get into a network administrator’s role, especially at a smaller company. But at a larger one you’d most likely have to start in a more junior position and make yourself ready to move up as openings become available.
Knowing how to apply the technology to improve the company’s product, process or profit. That’s the key.
Best of luck,
Maxwell