The breakdown of finite element (FEM) computations for the steady symmetric two-dimensional flow of dilute and ultradilute Oldroyd-B fluids around a cylinder in a channel, at Weissenberg numbers Wi= O(1), is shown to arise due to a coil-stretch transition experienced by polymer molecules in the wake of the cylinder in the vicinity of the location of the stress maximum on the centerline. In dilute Oldroyd-B fluids, due to the modification of the flow caused by the presence of the polymer, the coil-stretch transition leads to the stress maximum diverging toward infinity at a finite value of Wi approximate to 0.7. On the other hand, in ultradilute solutions, the stress maximum approaches infinity only as Wi ->infinity. In FENE-P fluids, the coil-stretch transition leads to the mean extension of the molecules saturating to a value close to the fully-extended length, with the maximum stress remaining bounded with increasing Wi. An estimation of the number of finite elements required to achieve convergence for ultradilute Oldroyd-B fluids reveals that obtaining solutions at Wi > 1 is not feasible. (C) 2008 The Society of Rheology.