An infinitely long elastic plate/layer is under uniaxial compression with its long dimension held by adhesion to a flat rigid base without friction. A prescribed length of the plate/layer is free of adhesion. This configuration is similar to a pre-stressed elastic film for which buckling of an unbonded section is a necessary, but not sufficient, condition for delamination. For that configuration, buckling occurs at the Euler buckling load of a fixed–fixed plate. Although the present study does not include friction or tangential interface stresses, the onset of buckling should be similar for these two cases. For the case of an elastic plate, a cohesive zone is used and it is found that the fixed–fixed buckling load is not attained except for extremely large values of a cohesive zone parameter. For realistic values, the buckling load is about half of that value. For the situation of an elastic layer with adhesion (without a cohesive zone), the buckling load approaches the fixed–fixed value only for very large values of the ratio of the unbonded length to the thickness.