The recently discovered reversible strain effect in Y-Ba-Cu-O (YBCO) coated conductors contrasts with the general understanding that the effect of strain on the critical-current density J(c) in practical high-temperature superconductors is determined only by crack formation in the ceramic component. Instead of having a constant Jc as a function of strain before an irreversible drop when cracks form in the superconductor, J(c) in YBCO coated conductors can decrease or increase reversibly with strain over a significant strain range up to an irreversible strain limit. This reversible effect is present in samples fabricated either with rolling-assisted biaxially textured Ni-W substrates or with ion-beam-assisted deposition on Hastalloy substrates. The reversibility of J(c) with strain is observed for thin as well as thick YBCO films, and at two very different temperatures (76 and 4 K). The reversible effect is dependent on temperature and magnetic field, thus indicatine, its intrinsic nature. We also report an enhancement of the irreversible strain limit epsilon(irr) where the reversible strain effect ends and YBCO cracking starts. The value of epsilon(irr) increases from about 0.4% to more than 0.5% when YBCO coated conductors are fabricated with an additional Cu protection layer.