A family name carries far more than the size of the wallet that accompanies it.
Class is all about breeding and the right "Names".
There is a fair amount of idiots due to the size of the gene pool that they are drawn from (although this is not so prevalent now thanks to the inroads the lower classes are making

).
But it is fair to say that the Upper classes get better schooling and this immediately puts them ahead of their rivals in the lower classes.
It is this difference that allows a stupid upper class person to do well (thanks to a name, school network and a little shoulder up from one of Daddy's friends in the city).
I'd say the I was Middle to Upper class and whilst am not the richest of people I know (only the Noveaux brag about their wealth and personal effects), my schooling and upbringing puts me into this social class.