I think we do!
Not as set in stone as it once was and not insurmountable for individuals who have the tenacity to see a way through and out the other side if born into a life of deprivation. It can be turned around in a generation, not impossible but a damn sight easier for the child born into a world if everything is in place to make their ascent into a successful life almost a foregone conclusion.
Nevertheless for some I imagine it would be impossible to move up the social ladder as so much is stacked against them, the vicious cycle of having a poor upbringing themselves and and then being so ground down by their own demons and poverty that to even parent their child/ren is a struggle let alone imbue them with aspirations, add failing schools, lack of facilities into that mix and odds are stacked against that child from birth.
Lack of opportunities and where you are born are deciding factors, for example, a child born in say Barrow in Furness where from what I read life is bleak to parents on the poverty line who have addictions is probably not going to fare anything like as well as one born into a comfortable home in Tunbridge Wells for example. where schools are good. The added bonus of having parents who in turn inspire their own offspring to work hard at school, engage in extra curricular activities and generally develop them as people is that child's good fortune.
Whilst everyone is equal, not everyone has equal opportunities, there are enormous disparities depending on where you happen to be born, who you're born to and the standard of schooling received and that in itself is a can of worms as to levelling up the playing field!