It's not as impossible as some people say, but it's just very unlikely. Over generations families tend to change. George Bush´s great great great grandfather was a very intellectual scholar. FDR's children were some involved in politics but weren't very charismatic or succesful in it when they didn't have their father's support.
If you take a look at royal families the reason democratic dynasties rarely hold out becomes obvious.
While the history of Europian monarchies may contain many smart and capable individuals they are also a good demonstration that it takes more than good genes and good teachers. These good genes may splice up differently and the lessons may be wasted. F.x. the difference between Frederick the great of Prussia and Wilhelm II. Or if we look at Denmark, where some of the kings in 1500s were smart politicians their descendants were some mad and some stupid. (And some were ok). Or, Napoleon, pretty smart guy, his nephew... Well, I think Napoleon III actually isn´t stupid, he was good at politicking but bad at geo-politicking.
It seems these things can only last two generations at most, if the grandfather and father are smart, odds are that the son is an idiot that is better of letting parliament rule in his stead. (The opposite can be true).
So, if Washington had a son who´s the spitting image of his father and easily wins an election in the 1820s... well I´m not placing a bet on his grandson. But I think having the last name, Washington, Jefferson, Lincoln or Roosevelt would be an asset. It´s just unlikely the man/woman born with it is the second coming.