There's no such thing as a 'British' accent, that would imply some kind of weird amalgam of generic English, Scottish, Welsh and Northern Irish accents. Okay, I think I know what you're alluding to, it seems when most people from the other side of the atlantic talk about 'British' accents they refer to the stereotypical elocutionary accents of the upper/middle class found in central London, that are oft portrayed for giggles in the mainstream American media. Perhaps to a lesser extent they also refer to generic English accents that are harder to pin down to any particular region, but basically you'll never see somebody listen to a Scottish or Welsh accent and opine that it's 'British'. If they would simply refer to the accent which they envision as an 'English' accent then I wouldn't really have a problem with it, but as it stands calling it 'British' seems rather disingenuous. And as others have mentioned nations encompass a wide variety of dialects; Jeremy Irons would likely seem to have an accent which as a great deal more sophisticated than say Wayne Rooney, whereas on the other side of the pond Kelsey Grammer's voice would sound (I'm not saying the guy himself is all that classy, as he isn't >.>) more refined than Bill Hicks. Personally, I come from a place where the stereotypical accent would sound something like this:
Though I myself (and many others I've met) haven't picked it up, leaving me with a rather generic English accent, that's rather difficult to trace, unless you can pick up on the slight Somerset inflection that my voice places on certain words but I wouldn't consider it particularly posh or anything. Ultimately I find this prospect to be rather pretentious, the sound of one's voice does not denote any particular qualities or characterstics about the individual themselves, it's only an indication of where they grew up and even then the strength of one's accent tends to vary from person to person. And I honestly have much less respect for someone trying to obfuscate their natural speaking voice for something more accessible in an effort to be taken more seriously than someone who is comfortable enough in who they are and their ability to articulate themselves in spite of whatever mypoic preconceptions that others may have about their voice. In short, what may constitute as sophisticated is entirely down to the individual in question and even then just because someone sounds sophisticated doesn't mean they actually are.