The terminal server has exceeded the maximum number of allowed connections error?

You can get this error when trying to connect to your VPS, VM, or dedicated server via Remote Desktop.  It can happen if there are too many other users logged in at the same time.  Most likely the person who was logged in before did not log out fully.

You will need to connect to the VPS through the console and log off the other remote desktop users. You can do that by opening up the Remote Desktop Connection window as normal but instead of putting in just the VPS IP address put in:

IPADDRESS /admin

On some older versions of remote desktop you will have to use /console instead of /admin

Once logged in you can log off the other users under:
Start >> Administrative Tools >> Terminal Services Manager

If you can't see this straight into the Start menu, go to Start >> Control Panel and you should see Administrative Tools there.

Right click the logged in users (bar yourself logged in to the console), and press "Log Off".

To avoid this problem in the future please always log out of the VPS when connected via Remote Desktop rather than quitting out of the remote desktop window itself.