I assume the T-Mobile Rocket laptop stick is a mobile wireless data access device?
From the symptoms you are describing, I would suspect wireless interference of some kind. Maybe a neighbor recently got a wireless system that is operating on the same channel as your equipment?
Does your T-Mobile Rocket device allow for different channel operating frequencies?
Did you explain to TiVo that you are trying to connect through a mobile carrier device and not a "regular" internet connection?
Have you tried contacting T-Mobile support to see if they might be blocking TiVo's IP address for some reason?
Edit: Do all of your other internet dependant functions (Netflix, Pandora, Amazon VOD, etc.) still work?
Edit #2: (Longshot) If these are the original Series 3 OLED Tivos (faceplate looks like a sideways thermometer), there's a slight possibility that the power supply might be showing signs of capacitor plaque
. Power supply issues can manifest in many different ways, more commonly from high draw devices like the hard drive. I would think it is rare that that could be your problem (as long as everything else works), but it wouldn't hurt for you to go ahead and eyeball the power supply capacitors. You might pre-empt further catastrophe.
Edit#3: Welcome to TiVo Community Forum!