Yes. See D.R. Lick's article, "Sets of non-uniform convergence of Taylor Series."
It is shown there that for every closed subset $F$ of the boundary $\partial D$, there is such a series that converges everywhere on the closed disk to a continuous function, and whose set of non-uniform convergence is $F$. This means that the series converges uniformly in a neighborhood (open arc) of each point in $\partial D\setminus F$, but not in any neighborhood of any point in $F$.
I have been interested in this question before, which is why I have a reference handy. I referred to this article in another answer, to a question about power series in Banach algebras. As mentioned there, the sequence of Cesàro means of the power series will converge uniformly on the closed disk.