Are there any molecules with the same spectrum?
In addition to the R/S chiral molecules mentioned by @user26143. There are also molecules with a distinct atomic composition that have experimentally indistinguishable spectra, even though theoretically there will be minor differences. One example, according to Rabitz et al. 2009 PRL (freely available for viewing) are flavin mononucleotide and riboflavine and in fact many other flavins. Due to there very similar structure you need unconventional techniques to be able to distinguish them.