I think the overqualified here probably is in respect to the points system. You gain points for the level of education that you have, what your nationality is etc. For example, I've got points to burn here being an EU national with a bevy of academic qualifications.
We are, 'though for staying for longer than three months you need to have a job or enough resources such that you won't "become a burden on the social services of the host Member State during the stay".
just to expand, it is highly unlikely you would have any issues with the three months period in general, as there is no structure to actually check if you have been in a country more than three months or not (no borders so you can most likely claim that you were in another country until yesterday).
I'm joint EU and Australian - so that's probably why I had the points system in my head. There's no points system for the first three months. There are, however, restrictions on EU citizens after three months, as you need a registration certificate. As far as I can see, you can get this if you're either employed, able to fend for yourself while studying, or are filthy rich.
