By adding minute concentrations of a high molecular weight polymer, liquid jets or bridges collapsing under the action of surface tension develop a characteristic shape of uniform threads connecting spherical fluid drops. In this paper, high-precision measurements of this beads-on-string structure are combined with a theoretical analysis of the limiting case of large polymer relaxation times, for which the evolution can be divided into two distinct regimes. This excludes the very late stages of the evolution, for which the polymers have become fully stretched. For times smaller than the polymer relaxation time, over which the beads-on-string structure develops, we give a simplified local description, which still contains the full complexity of the problem. At times much larger than the relaxation time, we show that the solution consists of exponentially thinning threads connecting almost spherical drops. Both experiment and theoretical analysis of a one-dimensional model equation reveal a self-similar structure of the corner where a thread is attached to the neighbouring drops.