Is there a command that I can run in-game to open up more slots during a networked game of Quake 3 Arena?
Add this line to your
seta sv_maxclients 16
sv_maxclients 16 - maximum number of players on the server.
You can try using rcon to change this while the server is running:
/rcon seta sv_maxclients 16
..but even if that is possible normally, most hosting companies disallow that because you pay by the number of server-slots.