YBa2Cu3O7-delta (YBCO) films with thicknesses ranging from 1.0 to 6.4 mu m were deposited by pulsed laser deposition on rolling-assisted biaxially textured substrates (RABiTS). The RABiTS were of the configuration CeO2/YSZ/Y2O3/Ni-3 at.% W. As the YBCO film thickness increased, I-c continued to increase and reached similar to 300 A/cm width for a 4.3 mu m-thick YBCO film. Commonly observed mechanisms for J(c) decrease with increasing YBCO film thickness were not observed. Homogeneous microstructures obtained in even the thickest YBCO films, suggest that the I-c/width can still be enhanced considerably.