We investigate the stability of a polymer thread imbedded in a matrix that is confined between two parallel plates. Utilizing a combination of experiments, numerical simulations (lattice-Boltzmann), and surface area calculations, we find substantial deviations from the classical results when the diameter of the thread (Do) is comparable to the height (H) of the matrix. We find three regimes as a function of H/D-0: For H/D-0 greater than or similar to 3, the thread breaks up into droplets through a finite wavelength axisymmetric capillary instability as described by Rayleigh and Tomotika. For 1.3 less than or similar to H/D-0 less than or similar to 3, the effects of the confinement are felt; the shape becomes nonaxisymmetric, the early-stage growth rate decreases, and the wavelength increases. For sufficiently low H/D-0, we observe that the thread is stable with respect to the capillary instability over the experimental time scales. The simulations qualitatively agree with the experiments and reveal that while the shape of the growing bulges is nonaxisymmetic, the narrowing necks are circular. A simple surface area consideration then shows that as the wall-induced asymmetry of the fluctuation increases, the minimally unstable wavelength increases and eventually diverges.