How are you connecting the 360 to the router? Ethernet or wireless?
If wireless, does the 360 have the correct WPA key & is it on the router's list of approved MAC addresses (if enabled)?
Whether you use ethernet or wireless to connect the 360 to the router, you do not need to use a static IP. Just set everything on the 360's network settings to automatic.
Regarding connecting the 360 directly to the cable modem, if you do that you need to power cycle (reboot) the modem, so that it can connect to a different device.