We considered a polycrystalline cylindrical wire of the initial radius R0 composed of identical cylindrical grains of the length L0, strained uniaxially by an external stress P. At the temperatures at which some surface and grain boundary diffusion are allowed the thinning of the nanowire in the vicinity of grain boundaries occurs due to the phenomenon of grain boundary grooving. We calculated the equilibrium shapes of the nanowire achieved after long annealing times. Our calculations demonstrated that for any given L0/R0 ratio some critical value of the applied stress exists above which the nanowire is unstable and breaks down into the string of isolated spherical particles, in full analogy with the Rayleigh instability of long cylinders. The kinetics of the shape change was calculated numerically. It was shown that the rate of thinning of unstable wires diverges as the moment of breakdown is approached. We also demonstrated that the breakdown may occur even for nominally stable wires “on the way” to achieving their equilibrium shape. Therefore, the stability of nanowire is determined by a combination of geometric (L0/R0), thermodynamic (grain boundary energy), and kinetic (ratio of grain boundary and surface diffusivities) parameters. An application of external tensile stress accelerates the breakdown of nanowires.