In this paper, we address the accuracy of a posteriori error estimators for finite element solutions of problems with high orthotropy especially for cases where rather coarse meshes are used, which are often encountered in engineering computations. We present sample computations which indicate lack of robustness of all standard residual estimators with respect to high orthotropy. The main culprit is the coarseness of the finite element meshes relative to the thickness of the boundary and interface layers in the solution. This is often imposed by the size of the problem domain, the capabilities for mesh generation, and the employed finite element software and cannot be changed. Hence, there is need for a posteriori error estimation approaches which are robust for coarse meshes and which are constructed using non-intrusive approaches with respect to the employed software.