Hi JB,
Strange you've had no replies on this, but I'll post my findings.
I have tried to get BBC Radio 1 and BBC Radio 2 and they play fine listed as wma 48kbs as you found.
I cannot, however get the BBC HiDef test channels to work, so vice versa to you.
I do, however, get a reliable stream of Radio Paradise at AAC 320 Kbps, and other higher rate bit streams, so I know it's not a problem with my broadband capacity.
I think there are a couple of reasons why some people can get a station and others not, not withstanding their own broadband's ability to stream higher bit rates, of course.
It would seem that Internet radio servers only have a certain capacity to stream and if there are a lot of people listening at any one time, you may struggle to get connected. Secondly, depending on where you are in the world, this would appear to affect your ability to receive certain stations.
I think the bottom line is that as long as your HDX is receiving some channels at various bit rates and formats, it's most likely the issue you cannot receive certain stations, lies elsewhere than your HDX. The question as to why you can't get certain BBC stations is any ones guess! Try again at another time and another day, you may be more successful.
I'm not an expert by any means, but in lack of any other replies here, that's my take on it! The more informed here, may be able to help you further. In the mean time, try powering your HDX and router down and up again and see if this solves the issue if you are still not getting connected today.
All the best