What defect is inherent in a purely classical education?
I assume that by classical education you mean a type of literary education that is based on a curriculum devoted to the authors of the Greek and Latin Classicism with maybe some major canonical western writers. This idea of education remains confined in the literary area and runs the risk of downplaying the importance of technical and more practical skills in the education of an individual. This attitude has caused the assumption that a humanist type of education is somewhat superior to scientific education and that it is the true knowledge because it has no practical application. In La Tête bien faite, French philosopher Edgar Morin has powerfully argued against this rigid separation of disciplines, advocating a unitary vision of knowledge against the tower of Babel of the different disciplines. Quoting Montaigne, Morin prefers a "well-made head to a well-filled head".
In a broader sense a Classical Education would be the liberal arts ideal with perhaps a major in Classics. But this might also include courses in the history and philosophy of science as well as the usual core distribution in sciences, social science and humanities. Even in the narrower sense provided above, such a student would be well prepared for a second BA and MA--as often happens now--in Physics or Biology. It is the learning how to learn that counts--unless she or he somehow persists in believing it was all downhill after Aristotle--an unlikely outcome for anyone educated in the modern world in any fashion.