Your own broadband speed has nothing to do with the speed you get through your Tivo box. It is limited to a 10Mb connection and should be enough. Sounds like you are feeling the same issues I had for years until very recently. That is that your area is known for high utilization at times and they are working on upgrading the network in your area. They give you a proposed date and then every time you check that fault with them the date moves.
Mine does seem to have eventually been sorted but it has been over a year before I could watch iPlayer in HD without any sort of crashing or buffering etc. Ridiculous from a telecoms/tv firm when I can watch it on my phone out in the street perfectly.