The English will correct your accent, by lightly making fun of it. It is most clear when you are an immigrant from an english speaking country or switch class brackets.
I assume they are talking about moving from "working class" to "middle class" (N.B. It's not really possible to move from "middle class" to "upper class" unless by marriage and then it's only really your kids that would move class bracket.
In England, accent/grammer usage is a strong indicator of class (as I assume it may be in other countries).
Basically what MrsPeaches said and with more stratifications. I see way more than 3 brackets, the accent groups and all the other trappings of class stretch from the mythical heavenly top, people you can only theorize to exist all the way down to missing teeth and barely speaking english in hell underneath the poverty line.
I've been lucky enough to see a lot of the levels, it's vastly different to my home country which only really has the three levels (it isnt discussed) and is largely culturally homegenous.
