A couple of years ago I hooked my A1200 up to the Internet using Miami DX and suffered issues. The solution I was given on Amiga.org (which worked for me) was to add the DNS server addresses into my router manually (varies from router to router).
However, you have also tested this with the cable modem straight into the Amiga, ruling out DNS config issues - when you plug your cable modem into the Amiga, do you reset it by powering it off for 10 seconds? I am with Virgin Media and if the modem gets plugged into any other device, it has to be reset to assign the IP address to the new MAC address it is communicating with.
