In this article, we numerically study the regularity loss of the solutions of non-parametric minimal surfaces with non-zero boundary conditions. Parts of the boundaries have non-positive mean curvature. As expected from theoretical results in such geometry, we find that the solutions may or may not satisfy the boundary conditions depending upon the data. Firstly, we validate the numerical study on the astroid and discuss the various kinds of non-regularity characterizations. We provide an algorithm to test the regularity loss using the numerical results. Secondly, we give a numerical estimate of the threshold value of the boundary condition beyond which no regular solution exists. More theoretical results are also given on the approximation by the regularized solution of the non-regularized one. The regularized solution exhibits a boundary layer. Finally, the study is applied to the catenoid for which the exact threshold value is known. The exact value and the computed one are in good agreement.