My university tells me I can't use a router in my dorm. Is there any way they can tell?

Solution 1:

Adding to PulpSpy's (correct) answer, it's also possible to detect routers (NAT or not) by looking at the TTL field of outgoing IP packets. End stations usually set TTL to a known number, such as 64, 254, or a few other alternatives depending on the OS. When a majority of packets are one less than this, such as 63 and so on, it indicates there was a router hop in between.

Solution 2:

Yes they probably can tell. NAT routing will reassign all the port numbers to keep which traffic originated from which computer straight. As a result your traffic will look strange and when more than one computer are connected, they will typically be on adjacent ports. It wouldn't be proof, but enough to attract attention if they were specifically screening for it.

Solution 3:

In addition to the TTL already mentioned in the other answers, they could be using the DHCP fingerprint of your router when it'll obtain an IP from it's WAN port.

I know this because I work on the PacketFence open source NAC (a Cisco NAC competitor) and we use such tricks.

Here's the list of recognized DHCP fingerprints in PacketFence: http://packetfence.org/dhcp_fingerprints.conf

We know it's also used by other products.