Serverschool.com showed me just the command I needed to find out the number of users on a Linux Server.

It goes like this:

cat /etc/passwd | wc -l

This outputs a single number that indicates the total number of users on the system - including "non human users" such as the users  created by the system for purposes of running daemons.

I ran the above command on my newly acquired Brainhost account and it revealed the total number of users on the server to be 1590.