From a slightly different angle from the other replies: the MAC address is Ethernet-specific. It's how two Ethernet cards on the same network identify and talk directly to each other. Each Ethernet card, worldwide, is supposed to have a unique MAC address, a prefix identifying the vendor, and then a unique additional code. In practice, there are probably some repeats, but not very many. This lets almost any two cards in the whole world plug into the same network segment and expect things to work; a MAC has to be unique on the local network segment. (Otherwise, the cards would get confused about who was talking to whom.)

IP addresses run on top of that. MACs are specific to Ethernet; IP addresses can work on many different network types. On most networks, there will be a machine called a DHCP server, which hands out temporary IP addresses for people to use. It logs that information semi-permanently, recording which MAC had which IP at what time.

Because MACs are globally unique, this is a very strong indicator that your specific network card was the one to have a particular IP.

However, most network cards will default to their hardware ID, but can be configured to use any arbitrary address. If you do this early enough in the boot process, the real hardware ID may never be used on the network at all.

So, in this case, the university had records indicating that, when the alleged copyright infringements occurred, the IP was linked to a specific MAC. (probably his real, physical router ID.) So they search the network to see who has that MAC, and it's ... nobody. It's gone, not on the network anymore. (Networks can log which MAC addresses are on which ports, but very few of them do.) Poof, the perpetrator has vanished!

If they raided every room and inspected every device with an Ethernet port, they could figure out which router actually had the ID that was infringing. But in the US, at least, they'd need probable cause to do that.