Has no one taken a network capture on a failing PC using something like Wireshark?
In my experience XP and W7 can have a similar problem occasionally. Unfortunately W7 apparently no longer has the click option simply to update the lease.
One problem seen in the past was when a user's laptop moved between locations. The PC did its DHCP request before its network layer was full operational. As it did not see any answer to its DHCP request - then it used the previous gateway address. That was not valid so the PC assigned itself a Microsoft default IP address - and then needed a manual prod to try DHCP again.