In layered superconductors (SC) with small interlayer Josephson coupling vortex-antivortex phase fluctuations characteristic of quasi two-dimensional (2D) Kosterlitz-Thouless behavior are expected to be observable at some energy scale Td. While in the 2D case Td is uniquely identified by the KT temperature TKT where the universal value of the superfluid density is reached, we show that in a generic anisotropic 3D system Td is controlled by the vortex-core energy, and can be significantly larger than the 2D scale TKT. These results are discussed in relation to recent experiments in cuprates, which represent a typical experimental realization of layered anisotropic SC.