Data on the Tl-205 NMR frequency shift and linewidth due to the magnet
ic field variation in the mixed state of Tl2Ba2Ca2Cu3O10 are presented
. These data, which are commonly used to extract the London penetratio
n depth, show a near linear temperature dependence at T << T(c). We of
fer an explanation of this effect as being caused by thermal fluctuati
ons of vortices in weakly coupled layered systems. We argue that the f
luctuations cannot be ignored in interpretation of NMR and muon-spin-r
esonance data for materials with large penetration depths even at low
T.