Droplets containing a dilute polymer solution enter a T-shaped microfluidic junction and stretch as they pass through the stagnation point. Depending on the initial aspect ratio and speed, droplets may break into two segments. We characterize the breaking-nonbreaking behavior of these droplets and find that viscoelastic droplets are less stable than Newtonian droplets of comparable shear viscosity. When droplets break, we observe that viscoelastic droplet segments are connected by persistent filaments that first undergo stretching due to drag of the outer viscous liquid on the large connected segments. Later, filaments undergo iterated stretching and develop a series of beads along their length. Secondary filaments between beads no longer stretch with the outer flow, but rather they exhibit an exponentially decreasing diameter consistent with elastocapillary breakup. The relaxation time obtained from the filament diameter profiles is close to the estimated Zimm relaxation time for the polymer solution, but depends on the global flow parameters, including the initial size and speed of the droplet prior to entering the T-junction. Based on the microscale diameters of the filaments, we suggest that splitting of droplets at microfluidic T-junctions may be a useful way to characterize the extensional rheology of low viscosity elastic liquids. (C) 2009 The Society of Rheology. [DOI: 10.1122/1.3086871]