Zadoff-Chu (ZC) sequences have been used as synchronization sequences in modern wireless communication systems, replacing the conventional pseudo-random noise (PN) sequences, due to their perfect autocorrelation properties. This paper shows how frequency offset can damage the autocorrelation of the ZC sequence, causing irreducible timing errors. An analytical framework is developed, which characterizes a ZC sequence's timing behavior and its fundamental limitation as a synchronization sequence in the presence of frequency offset between the transmitter and the receiver.