The two are not linked all the time but they might go together a lot of the time iyswim. I'm judging this from a bit of an outside perspective as I'm not British and the whole class issue seems very Britain specific to me.
For example, DH's family, the maternal side, is working class through and through. In the case of his grandparents, it's in a good way. They are grounded, have always worked for the little they had and are genuinely nice.
Then you get to the generation of their children. DH's aunt and uncle. They are what you would call "working class done good". His aunt has no qualifications but married a fisherman with his own boat. They have 2 small children and plenty of money. A massive house, expensive cars, loads of expensive holidays. BUT they don't have the common sense, intellect or whatever you want to call it, to go with the money. While the children are only dressed in the most expensive clothes, they don't own a single book. Never had picture books or the like as toddlers but got computer games, own dvd players etc at the age of 2! An Ipad and mobile phone at 3! I kid you not! For the his 4th birthday, one of the boys got a quad bike, not a kiddy play one but a proper one, to drive around in the field behind their house. DH tried it out, the thing is heavy, big and super fast for a child that age, 40mph or something like that. It's not a toy, it's a fatal accident waiting to happen but his aunt and uncle, who are genuinely nice people btw, believe that the more something costs, the better it is as a gift. Educational value etc is not taken into consideration at all. The children are also only fed garlic bread, chips and chicken nuggets because that's what they ask for.
Sorry if this is all judmental of me but it's how I see it. Hope my explanation makes sense.