We report on measurements of the low temperature dependence of the magnetic penetration depth λab in several single crystals of La2-xSrxCuO4, Bi2Sr2CaCu2O8 and YBa2Cu3O6+x at various doping levels ranging from the under- to the overdoped regimes by using a novel single-coil technique achieving 10 pm resolution. We have found a linear dependence of λab in all samples with a rapidly increasing slope dλab/dT as doping decreases. Our analysis of the data indicates that a superconducting gap with d-wave symmetry is sufficient to quantitatively account for the above slope values in the optimally or over- doped samples. In the underdoped samples, the d-wave model predicts much smaller values than those measured by assuming realistic values for the zero-temperature λab and gap Δ. The experimental data are compatible with a model of thermodynamic phase fluctuations of the order parameter. Therefore, we put forward the hypothesis that the gapless properties observed in cuprates may have qualitatively different physical origins depending on the doping level.