Can One Learn History from the Allelic Spectrum?

It is well known that the neutral allelic frequency spectrum of a population is affected by the history of population size. A number of authors have used this fact to infer history given observed allele frequency data. We ask whether perfect information concerning the spectrum allows precise recovery of the history, and with an explicit example show that the answer is in the negative. This implies some limitations on how informative allelic spectra can be.