Mini cable box

I just got off the phone with a service agent who told me that the mini cable boxes that we now need to view non-DVR connected TVs do not work on a splitter. All of our connections are used for TVs including the one I use to connect my Netgear data gateway.

I was not told nor can I find information that explains that the mini boxes must be on a dedicated cable connection. Now I will have to forego a TV in one of the bedrooms so that I can continue to use the internet.