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Let $u$ be harmonic in the half-space $R_{+}^{n+1}$, $n\ge 2$. We show that $u$ can have a fine limit at almost every point of the unit cubs in $R^n=\partial R_{+}^{n+1}$ but fail to have a nontangential limit at any point of the cube. The method is probabilistic and utilizes the equivalence between conditional Brownian motion limits and fine limits at the boundary. In $R_{+}^2$ it is known that the Hardy classes $H^p$, $0\<p\<\infty$, may be described in terms of the integrability of the nontangential maximal function, or, alternatively, in terms of the...