By investigating the relation between growth and value distribution A. Melas established a qualitative version of a Picard type theorem for universal Taylor series, that is: every universal Taylor series on the open unit disk $D$, assumes every complex value with at most one exception on infinite subsets of $D$ that approach the boundary of $D$ rather slowly. On the other hand, we show that there are universal Taylor series on $D$ such that the infinite subset of $D$ on which exactly one value is assumed, can approach the boundary of $D$ arbitrarily fast. Hence in view of Melas' work our result is the best possible. We also study the problem of interpolation by universal Taylor series.