We report the results of systematic measurements of the decay of persistent currents of superfluid 4He films as a function of both temperature and film thickness. For film currents where the fractional decay per decade is less than 20%, the data can be represented by v=v0[1-ξ×ln(t/t0)]. For stronger decays we document deviations from the lnt behavior and observe that the empirical rule v=A(B+t)-n reasonably represents the data over the full range of our studies. The Iordanskii-Langer-Fisher thermal fluctuation theory is examined in the vortex pair model, and inadequacies of the theory are discussed. A comparison is also made to the predictions of Donnelly, Hills, and Roberts.