Something else to consider:
FEM can only give you stresses, strains, eigenvalues, etc. But it does not tell you what to do with these results. Even if the FEM is done correctly, the results can easily be misinterpreted.
For example, consider the local stresses around a hole. For a ductile material (typical aerospace metallic), the static capability is not significantly affected by the local peak stresses (not addressing fatigue at this point). However, I have seen structure sized (incorrectly) to these local stresses. Nothing was wrong with the FEM.
Another example is for a composite material. What do you do with the local stresses? If you use the peak, you may be too conservative. But you must consider the local effects as well, just not perhaps the peak stress (i.e. characteristic dimension approach). No FEM can tell you that. It can only report the stresses. If you were not familiar with the point stress criterion (or the like), you may not produce an acceptable result.
There are problems that rely on semi-empirical models (test data) to be accurate. Since the test data is integral to the mathematical formulation, a FEM usually cannot capture these effects. Usually semi-empirical approaches indicate a problem where the theoretical solutions do not yield satisfactory engineering results. So by the very nature of the problem, the FEM (on its own), won't properly solve the problem.
Without good understanding of the classical solutions, you would likely fail at producing an acceptable result for these scenarios. I can think of many more, but I don't want to bore everyone.
Brian