Two-dimensional (2D) barrier materials such as graphene, boron nitride, and molybdenum disulfide hold great promise for important applications such as DNA sequencing, desalination, and biomolecular sensing. The 2D materials commonly span pores through an insulating membrane, and electrical fields are applied to drive cross-barrier transport of charged solvated species. While the low-voltage transmembrane transport is well-understood and controllable, high-voltage phenomena are uncontrolled and result in the apparent breakdown of the 2D material's critical insulating properties. Here we use suspended graphene over a 50 nm silicon nitride nanopore as a model system and show that delamination of the 2D material occurs at higher voltages and can directly cause a number of the puzzling high-voltage transport observations. We confirm the occurrence of delamination and observe via atomic force microscopy measurement a micron-scale delaminated patch in a system using chemical vapor deposition graphene. Furthermore, we show that the conductivity of the same system is strongly correlated to the area of delamination via coincident current measurements and optical imaging of the delaminated area. Finally, we demonstrate that delamination alone can cause a dramatic breakdown of barrier function through observation of a reversible increase in conductance of samples prepared with pristine defect-free graphene. These findings should have a great impact on the design and interpretation of 2D barrier material for both experiments and applications.