We're behind some kind of Cisco routers, and have to connect via the Cisco NAC Agent. The tech support here at my university tell me that if I connect a router "the whole building will lose internet access." I find this really hard to believe, and I'd like to know: 1. Can they tell if I'm using a router and 2. How could they even tell if the router is using NAT?
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.
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.
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.
They can tell if your device is a router if it broadcasts routing packets (RIP, OSPF), and by the MAC address of the external ethernet interface.
Any interface that is broadcasting it's a Cisco device (or other router vendor) will attract attention.
Have fun learning, and try not to get suspended in the process!
I find it hard to believe that the network architects at a university would leave themselves vulnerable to a TOTAL network failure, in the event that a student jacks a router into his dorm room socket. The chance that a student might try such a thing is pretty high (witness this post, for example).
The Occam's razor answer, is that the tech-support guy probably was just trying to scare you, rather than say he just didn't know.
What's more, in a peer-to-peer architecture (unlike, say, SNA where you CAN trash the whole network by pretending to be the NCP), the likelihood of something like a device conflict causing complete network failure is pretty low. It COULD BE that your router would somehow be given (or you define it yourself) the SAME ip address of one of the dorm building's own distribution hubs, but it also COULD BE that an asteroid would hit your building.
But let's assume it does happen: as long as the cisco router doesn't get rebooted, nothing happens, and your router just complains that it can't get a proper connection. Also, the cisco router would probably report the duplicate IP collision in a log somewhere (which operators or network admins would notice).