Haven't had any issues myself but I have a thought...
The IP conflict is almost certainly the issue, when you reset the router the IP pool is reset as well. Pretty much all routers have a default config to use DHCP and they will set a IP range of over 200 addresses.
Unless you have changed the router settings I'm leaning towards yourPC as the cause.
The static IP that your ISP said you did not have was the wan or internet IP. The laptop could still be connecting with a static IP.
Two options to try to check that assuming you are in windows.
The first is to go to the control panel and pull up your wireless network connection status window. There is an option in status on one of the 2 tabs, should say details. You will be able to see if DHCP is enabled.
The shortcut is to just reset the TCP protocols. This will make the PC set all it's settings to run of DHCP, catch is if you are using assigned DNS or IP for something those will be lost (can't imagine why the PC would need a static IP.
From a command prompt (if in vista or 7 open as administrator) and run the following.
Code:
netsh i i r r
netsh winsock reset
Than restart the PC, I have to at least mention there is a chance that it won't reconnect to the wireless after that. If so just go through the windows wireless connection steps (select network, enter key) and it should be back.
Theory here is that the PC had a static IP set for a different network using the same IP ranges/gateway settings (more common than you might think).
Good luck!