Often when watching a movie ot using catch up it cuts out with a message that it cannot continue try later. With a movie it is usually after about an hour of viewing. Catch up TV especially BBC iPlayer takes a long time to start and runs in short bursts.
Loading apps on the TiVo box (which includes the iPlayer) is well known to be slow, it's the nature of the TiVo being quite old hardware these days and if you're using apps a lot - I would strongly recommend considering a swap to the V6 box if you also have VM broadband.
As for the streams stopping, this could be indicative of the box losing its internet connection. On the front-left of the box there's a series of green lights - the second from left has a heart symbol next to it, and is the "online"/connected light. This should be solidly on, but does it flash at any point?
If it does, that's indicative of a fault. Either with the box, or your local network.
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