The capillary thinning of a filament of viscoelastic liquid, which is the basis of a microrheometer, is analyzed in terms of a multi-mode FENE fluid. After a short time of viscous adjustment, the stress becomes dominated by the elastic contribution and the strain-rate takes on a value equal to two-thirds the rate at which the stress would relax at fixed strain. This strain-rate decreases as the dominant mode changes. At late times, modes reach their finite extension limit. The fluid then behaves like a suspension of rigid rods with a large extensional viscosity, and the liquid filament breaks. Predictions are compared with the experiments of Liang and Mackley (1994).