Thomas Sowell has a fantastic column pointing out that often the politicians who pose as intellectuals are, in fact, poseurs. Meanwhile, historically speaking, some of the more humble (and less formally educated) politicians -- like Harry Truman -- were actually very well-read.
As Sowell points out:
"As for reading the classics, President Harry Truman, whom no one thought of as an intellectual, was a voracious reader of heavyweight stuff like Thucydides and read Cicero in the original Latin. When Chief Justice Carl Vinson quoted in Latin, Truman was able to correct him."
... Echoing Sowell, Dan Flynn makes a good argument that, "Evidence of intellectualism for intellectuals has little to do with the intellect."