It doesn't sound like it would be a big deal, but Windows (and Mac OS X, I've since discovered, mid-7 minute `svn commit'...) just drop the connection if there's an IP conflict. That's actually quite annoying, at the very least. You have to resubmit commits, restart getting latests, your FTP craps out halfway through, file copies go wrong, you lose your remote desktop connection, the code finishes compiling before your web page loads, etc. With any luck, you won't lose any work.
If you don't run your code on the same computer you write it on, you might be less lucky. The debugger craps out and you lose your place, or the file server goes wrong and your code gets an error and stops or whatever while you were in the middle of waiting for it to get somewhere specific. And if you use any home-written tools, they're probably going to be even less resistant to network failures of this nature than the stuff you paid for - which doesn't do a fantastic job in the first place. It all adds up.
Fortunately for me, I wasn't using Remote Desktop all that much, and the device I was working with attached via USB. And I didn't lose any work when the source control just stopped mid-operation. And I could just put headphones on, and block out my colleagues' cries of pain. So maybe that's OK then, and it's no big deal?
I'd like to be able to say what the eventual solution to the problem was, but I don't remember.
When iPhone OS first came out, there was a bit of an uproar on many university campus networks that they were seriously screwing up the behavior of DHCP on networks with a very large number of transient clients.
How often do computers send out ARP requests to check on the uniqueness of their IP address? If it's less often than once per second or so (and I really hope it is), I could see how you could get away with having two devices with the same IP for a second here or there with no one the wiser.