The finite element method may converge in situations where the interpolation error does not; in particular meshes violating the maximal angle condition were discussed in the literature recently. However, there is also an example where the finite element method does not converge to the exact solution of the problem on these types of meshes. With a numerical study we stress that the finite element method converges; it converges to a function which is not the exact solution. A carefully designed a posteriori error estimator indicates this behavior. But a numerical test with the error measure as the difference to a solution on a very fine mesh of the same family leads to completely wrong conclusions: one observes a convergence order.