  We show that the cascade limit on ultra high energy cosmic neutrino
(UHEC$\nu$) flux imposes a lower bound on the neutrino mass provided that
super-GZK events of ultra high energy cosmic rays (UHECRs) are produced from
Z-bursts. Based on the data from HiRes and AGASA, the obtained neutrino mass
lower bound violates its existing cosmological upper bound. We conclude that
the Z-bursts cannot be the dominant source for the observed super-GZK UHECR
events. This is consistent with the recent ANITA-lite data.
