Summary: F. Móricz, F. Schipp and W. R. Wade [Trans. Am. Math. Soc. 329, 131–140 (1992; Zbl 0795.42016)] proved the a.e. convergence of the double \((C,1)\) means of the Walsh-Fourier series \(\sigma_n f\to f (\min (n_1,n_2)\to \infty\), \(n=(n_1,n_2\in \mathbb N^2)\) for functions in \(L \log^+ L ([0,1)^2)\). This result for bounded Vilenkin groups is generalized by F. Weisz [in: Approximation Theory and Function Series, Bolyai Soc. Math. Stud. 5, 353–367 (1996; Zbl 0866.42019)]. We show that these results can not be improved with respect to two-dimensional bounded Vilenkin groups (not only the two-dimensional Walsh group). We prove that for all measurable functions \(\delta : [0,+\infty)\to [0,+\infty),\lim_{t \to \infty}\delta(t) = 0\), \(G_m \times G_{\tilde m}\) two-dimensional bounded Vilenkin group we have an \(f \in L \log^+ L\delta (L)\) such that \(\sigma_n f\) does not converge to \(f\) a.e. (in the Pringsheim sense).