I am having about the exact same issue with the same product - 3TB My Book Live. I installed it just 2 days ago and I noticed some connectivity issues with my FIOS modem/router. After I reset it everything seemed normal - I was able to transfer files to / from the drive. When I got home from work yesterday, I was told the we had no internet connectivitiy. I went through the process of resetting the modem/router and got things working again. A couple hours later, the same issue occurred. I finally resorted to disconnecting the drive and have had no issues since. I should mention that I have very rarely had any issue with the modem/router and it has been fine for weeks at a time. It is almost as if the Drive is flooding the router with requests - my on private internal DDOS attack.
I will mention that the lights on the MBL seem to blink contantly - is that normal?
I will check to see if the MBL is set for DHCP.
Any help would be appreciated.