Data on the 205 T1 NMR frequency shift and linewidth due to the magnetic field variation in the mixed state of Tl2Ba2Ca2Cu30io are presented. These data, which are commonly used to extract the London penetration depth, show a near linear temperature dependence at T < T c . We offer an explanation of this effect as being caused by thermal fluctuations of vortices in weakly coupled layered systems. We argue that the fluctuations cannot be ignored in interpretation of NMR and muon-spin-resonance data for materials with large penetration depths even at low T. PACS numbers: 74.25.Nf, 74.40.+k In weakly coupled layered superconductors, thermal fluctuations of vortices strongly influence the thermodynamics of the system in a magnetic field. This has been observed in "non-mean-field" dependence of the magnetization M on temperature T and magnetic field H in Bi 2 Sr 2 CaCu 2 0 8 (Bi-2212) and Bi 2 Sr2Ca 2 Cu30io (Bi-2223) well under the mean-field transition temperature T C Q. At a temperature T*, a few kelvins below T c o, M is field independent [1-3], The effect has been explained quantitatively by the contribution of fluctuating vortices to the system entropy [4].For high anisotropies and H