BBC1,2 and 3 are all on the same broadcast multiplex (along with CBBC, and BBC news 24). ITV is on a different multiplex, along with channel 4, ITV2, ITV3, C4+1, ITV4, E4). BBC4 and Channel 5 are on two different multiplexes.
If the signal on the BBC1 multiplex is marginal it could be that the TVs tuner is just doing a better job of holding onto it, whilst the 870 is not. Still worth checking the signal strength on the 870 to see what it says - tune to freeview BBC1, then select System Menu/Initial Setup/Digital Tuner/Signal Check.
If signal check is showing OK, then it could be a faulty unit.