LOL I'm in a similar position to Moses. Although grammar schools were abolished by the time I received my education, the high school I attended was a 'former' grammar school and certainly maintained the standard. That and the fact it was out of town contributed greatly to a wide vocabulary and a neutral/slightly "posh" accent; certainly compared to my home town!
It's a strange experience to find oneself surrounded by people who sound like they're talking 'normally' to you (having grown up with the accent) but to be ridiculed by those same people for "t'funny way ye talk mate". Indeed. I also grew up on a council estate with a taxi driving father and a benefit claiming mother, but similarly, nobody believes me. Especially now I've moved away, attended university and settled in a "posh" town
Accents generally have no bearing on my perceptions of a person's intelligence, though it is fair to say that in my experience certain accents tend to belong to a greater percentage of less intelligent people than others

That's to be expected though, on the whole. Nobody can deny that some areas of the country are better developed, and have better (or worse) socio-economic standing than others. Despite what the government of the day might try to tell you, this standing or lack thereof will necessarily have a great impact on the resources (and standard of living) available in that area.
Focusing more on the actual OP's question, I have to say that I hate the Scouse accent. Sorry about that. Mostly because when we were made homeless years ago we were forced to relocate to Liverpool and the things we saw, and the treatment we received, were enough to put us of the people and the area for life. It's not all Cilla Black there, you know? LOL
Even the people who weren't openly hostile were only interested in drugs, thieving and treachery. Not to say all people in Liverpool are like that of course! Again it was more to do with the part of Liverpool we were in than Liverpool itself, but one's perceptions do tend to be coloured more by experience than theory. I should save myself somewhat by pointing out that I still have two very dear friends who are Scouse, so they're really not all bad!
While certain accents do sound homely (and I do like a good Lancashire or Yorkshire accent - I grew up in St Helens), it has to be said that they can't really form the basis of a true evaluation of a person's skillset. Generally I find that it takes less than a minute of someone starting to speak to evaluate their intelligence and social skills, and it has nothing to do with the accent they use to do it
