This varies from company to company. The best thing you can do is to get experience somewhere in the field. You could start out as a front line technician in a company. If you have a good work ethic, you might be able to ask your boss if you could work with the Network Administrator.
I am CISCO trained (no longer certified as CISCO certs run out after 3 years), and am A+ certified. I have about 15 months experience as a Computer Systems Administrator. I am currently studying for my Network+ certification. I also do not have a college degree, just two years of technical school.
Though I am not on the 'Network Operations Team' for my company (Lead by a network administrator), I find that my CISCO training is priceless in dealing with network issues, and I am able to talk to the network team very easily, as we are all on the same page with technology and terminology. I am also asked various network questions by my fellow systems administrators from my other company offices across the country.
As far as college degrees, I am not familiar with which one would serve you best. However, I do know that you will want at least four things on your resume if you wan to be a network administrator:
1. HANDS ON experience. This is vital. There are many problems that arise that a book simply cannot tell you about.
2. A+ Certification. This is a basic computer certification, and covers basic networking. You would be surprised how few technicians and networkers have this certification. It will help you stand out, as it not only guarantees a technical skill level, but also an ethical level as well (such as not by-passing production activation, not using pirated software, etc)
3. Network+ Certification. This is a basic, vendor neutral certification. This shows you have a broad range of networking knowledge: From cabling, to using Host Files, to IP Addressing, to file servers.
4. Actual instruction. Seriously. This one is commonly missed. Sure, some people learn networking from JUST being on the job and working along side of a network administrator. But, you never have the true 'from the ground up' knowledge that you should have. The worst thing you could have happen is for someone to ask a basic question (what is the max length of a cat5 cable run without being repeated?), and not know the answer. If you want to engineer, install, and maintain a network, you will need this. Only by having a firm ground work can you truly engineer an awesome network, and guarantee it will work as you say it will.
I hope this helps!!