Not true. Not all Presidents were wealthy (Most were) and not all Presidents inherited wealth (a substantial minority did). Some inherited wealth and were succesful in business prior to being President (Romney would be in that group if elected.).
Quite a few Presidents came from working class families (Obama, Clinton, Nixon, LBJ, Ike, Truman, Reagan from the modern era). Truman is the only President of the modern era that did not use his fame and influence, after serving, to make a lot of money. He steadfastly refused to do that and lived off his Presidential pension until his death. From the post war modern era only four (JFK, Ford, Carter, Bush I&II) came from wealthy families of which only JFK came for an ubber wealthy family.