How would you apply high voltage on a piece of dielectric making sure breakdown does not occur?
When an insulating dielectric is subjected to a high electric field, the electrons in the
valence band can overcome the energy gap and get excited to the conduction band. The mobile electrons are accelerated by electric field and so by collisions they excite more electrons to the conduction band resulting in an avalanche of conduction electrons.
This will happen in the presence of your electric field even if your electrodes are perfectly insulated with another material.  All it requires is the electric field strength at the dielectric to be greater than the breakdown value.
It's the field itself causing the breakdown, not conduction from the electrodes.