The EU only possesses the powers given to it through the treaties and - as a general principle - through the Member States. Powers not allocated to the EU remain with the nation states.
The important question is: Who decides if the Member States and EU institutions disagree? The answer is: The EU Court in Luxembourg. Over time, it has moved the borders to the extent that the Dutch EU Judge Koen Leonarts concluded that there is no nucleus of national competence which cannot be reached by the EU.
The future
This principle is repeated in the EU Constitution and is similar to the Swiss Constitution and to the 10th Amendment of the US Constitution (it has not prevented the USA from developing into a full federal state).
Links
See also Flexibility clause.