Hey, Sounds like Windows Server 2003 would be the ideal operating system in your situation.
When calculating the amount of connections the computer can handle there are a few limiting factors
1) The hardware inside the server (Memory, CPU, Disk Speed)
2) How many roles the server is holding (Domain Controller, Exchange Email, File Server, Print Server, DHCP, DNS, Etc would require more power)
3) Licensing. Microsoft server products are licensed either per user or per concurrent connection.
I would recommend Microsoft Windows Server 2003 Small Business Edition, just make sure you buy enough licenses to allow 15 computers to connect to it.
Your other option would be to buy or download a free copy of a Linux operating system, it would be compatible with your windows professional computers and would require less hardware resources. If all you are doing is hosting files in a central location, this would probably be the ideal. Remember linux can be a little trickier than windows, especially the initial installation. You may want to consider hiring an expert if it doesn't outweigh the cost of just purchasing a windows server.
I would not recommend installing the new operating system on the existing server. Convince your company to spend a couple thousand dollars and buy the server preloaded from Dell or another vendor. They may even offer an affordable service for them to come out and help you configure it.
Hope this was of assistance.