To explore the limits of layer wound (RE)Ba2Cu3O7-x (REBCO, RE = Rare Earth) coils in a high magnetic field environment > 30 T, a series of small insert coils have been built and characterized in background fields. One of the coils repeatedly reached 35.4 T using a single ~100 m length of REBCO tape wet wound with epoxy and nested in a 31 T background magnet. The coil was quenched safely several times without degradation. Contributing to the success of this coil was the introduction of a thin polyester film that surrounded the conductor. This approach introduces a weak circumferential plane in the coil pack that prevents conductor delamination that has caused degradation of several epoxy impregnated coils previously made by this and other groups. ; Comment: 7 pages, 3 figures, 1 table