Even if you had the theoretical victim's computer, intact with everything the same
as when he had it, any IP address you got using it would still lead to you unless
you also tapped his internet connection and used it too. The ISP is who controls
the IP addresses, not the hardware in the computer, and your window of
opportunity would be pretty small depending on how often the ISP sets leases to
expire and how quickly your victim hooks up his next computer to the same
internet account.

Actually having the hardware could make it possible to impersonate the other guy
but you would have to use the same physical connection and do it before he
registers a replacement computer with the ISP. In other words, you would have to
get close enough to be caught in order to try it.