Turn "Digital Captions" to "None" (Settings..Displays...Closed Captioning)
Leave "Analog Captions" alone..
Actually -- I found the opposite to be true -- turning "Analog Captions" to "NONE" and leaving "Digital Captions" at DTVCC1 worked.. My suggestion is to experiment with both settings and see what works best for you -- I do not use captioning on a regular basis, but it seems obvious that you do -- so you would be the better judge to see which combination of settings works best.
Thank you very much, Rick. It solves the closed caption problem for BBC America HD. Prior to this, I followed the directions from tivocommunity and turned off the digital closed caption, it works for every channel except BBC America HD.