We discuss the speculation that the sharp positron lines and correlated e/sup +/e/sup -/ and ..gamma gamma.. signals seen in heavy-ion collisions may be evidence for a new phase of QED. We examine several characteristics of the data which argue for this interpretation and point out further experimental observations which would favor this hypothesis. However, we detail also theoretical difficulties and experimental contradictions which considerably weaken the basis for this speculation. In particular, we argue that for the formation of a new phase or a solitonlike structure in QED it is necessary that nonlinear effects in electrodynamics become important. Even though Z..cap alpha.. is large, these effects always entail a suppression factor of ..cap alpha.., which is difficult to overcome.