The aim of the paper is to demonstrate the sharpness of theorems by F. Móricz, F. Schipp and W. R. Wade [Trans. Am. Math. Soc. 329, 131-140 (1992; Zbl 0795.42016)] on the a.e. convergence of the double \((C,1)\) means of the Walsh-Fourier series \(\sigma_n f\to f\) for functions \(f\in L\log^+L(I^2)\) where \(I^2\) is the unit square. The existence of a function \(f\in L^1(I^d)\) is proved such that slightly weaker conditions are satisfied (than the orignal conditions of the above theorems) and \(\sigma_n f\to f\) a.e. is not true.