By the use of symbolic dynamics, this note proves a result of B. Barna concerning real polynomials of degree at least 4 and having all distinct simple real roots. Specifically, the set of initial points, for which Newton's method fails to converge to a root of the given polynomial, is homeomorphic to a Cantor set. Also the note shows that the requirement for simple roots may be relaxed, and one still has Barna's result being valid.