To understand this needs a slightly broader understanding of NAT in general. The internet as a whole is fast running out of IPv4 addresses, and as a result, typically only your home router has a public IP. This router will then forward some or all ports to your PC / XBox / Playstation etc, which are all on private IP addresses. This means that other people external to your home network can connect direct to your gaming machine on these ports, rather than there being confusion as to whether they're connecting to other devices, such as your phone or tablet for example. Sometimes port forwarding has to be set up manually on a router, sometimes this is set automatically via uPnP. If all ports are forwarded to your gaming machine, then that is typically considered "Open / Type 1 NAT", because the machine is free to listen on any port. If the router only forwards a specific list of ports to the gaming machine, then that is known as "Moderate / Type 2 NAT". If you have multiple devices behind your router, perhaps in a shared house for example, you may start to find issues with your NAT type, as machines effectively fight for who the port is forwarded to. This would mean there will be devices that are unable to listen publicly for incoming connections. This is known as "Strict / Type 3 NAT".
At the University, this is no different. With 50+ ESports machines all behind the same router sharing a public IPv4 address, they can't all listen for incoming connections at once. The only way around this is to give each gaming PC its own publicly routable IP address, for which there are simply not enough available to do so.
In the new year, the University is looking to deploy IPv6 across its workstation estate. We have been allocated 1 Septillion public facing IPv6 addresses for use, which should more than cover the number of machines used for gaming! This won't be a solution for all games, as not all games support IPv6, but moving forward it should allow more and more games to run Open / Type 1 NAT at the University.