The drawback of only having one NIC is that it won't work, and it will completely hose your current network.That's not completely true. It's possible to run with only 1 nic if you set everything up correctly, I did for quite a while when my physical network setup wouldn't allow me the option. It can be done without completely hosing the current network. If you install on a box with only one nic the install will create and use a second virtual nic.
The default configuration will always make sure there are at least 2 NICs, even if it has to create a virtual adapter that shares the physical adapter. If you do no further configuration - for example, if do not understand the intricacies of how LinuxMCE configures the network and how to adjust your own network setup accordingly - then you will have 2 DHCP servers which will conflict - effectively hosing your network...
If you move forward with 1 NIC, and you're savvy enough to disable the DHCP on your router, you're still not out of the woods - since now you don't have a route to the internet - effectively hosing your network.
If you move forward with 1 NIC, disable the router's DHCP, and configure your Core's network setup just right (which takes at least some networking knowledge), then you've introduced an unnecessary point of failure into the setup, and you have an overcomplicated network for your home - but it does work ok. Of course, as soon as you tell anyone you have a 1 NIC setup you'll get flamed, because such a setup is "not supported".
2 out of 3 cases end in hosing your network. 1 requires a fairly significant amount of smarts - I'd guess that if one has enough smarts to get to that point, they would have read the FAQ and the Network Setup and immediately known how to get to that point. After that realization, one would immediately realize that such a setup doesn't really provide anything except extra work and complexity.
The bottom line is exactly what I said - put 2 NICs in the box (what's an extra NIC? $10?), save yourself a bunch of configuration work and unplanned downtime due to configuration mistakes or accidents or poor assumptions in the code, and avoid the flame war that this stupid topic always brings up. Anything is technically possible - it is Linux after all, and Linux is pretty good in the networking department - but the practical points outweigh the "it should be possible this way" points on this particular issue.
OP - Technically you can use only 1 nic, but don't. Take this from someone who has done it both ways.
Join the club.