You can change providers too and still watch all your recordings. I have moved TiVos between my two houses, one with CableOne and one with Time Warner Cable, and throw in some OTA, I can still watch every show recorded. The only catches are that some channels are copy protected and you cannot move those recordings from one TiVo to another (for archiving, or replacement). You can watch them or stream them to the mini's or other TiVos, you just cannot move the recording.
Broadcast TV channels should not be marked as protected on the cable system. If they are the cable company has made a mistake.