We study linear programming relaxations of Vertex Cover and Max Cut arising from repeated applications of the "lift-and-project" method of Lovasz and Schrijver starting from the standard linear programming relaxation. For Vertex Cover, Arora, Bollobas, Lovasz and Tourlakis prove that the integrality gap remains at least 2-ε after Ωε(log n) rounds, where n is the number of vertices, and Tourlakis proves that integrality gap remains at least 1.5-ε after Ωε((log n)2) rounds. We are not aware of previous work on Lovasz-Schrijver linear programming relaxations for Max Cut. We prove that the integrality gap of Vertex Cover remains at least 2-ε after Ωε(n) rounds, and that the integrality gap of Max Cut remains at most 1/2 + ε after Ωε(n) rounds. The result for Max Cut shows a gap between the approximation provided by linear versus semidefinite programmming relaxations.