Are both computers set to get their IP addresses automatically? If the first to boot gets an IP address by DHCP and the second has that same address configured manually, you will get the conflict.
Also, if you put a computer into hibernation, it won't have released its IP address. The other computer could be allocated that address by the router, which no longer sees the hibernated PC. When it comes out of hibernation, you have an IP conflict.
<<< Is there a tweak to resolve this or is it the nature of the beast? >>>
No, it isn't the nature of the beast and it shouldn't be happening.