  We re-analyze the compatibility of the claimed observation of neutrinoless
double beta decay ($0\nu\beta\beta$) in $^{76}$Ge with the new limits on the
half-life of $^{136}$Xe from EXO-200 and KamLAND-Zen. Including recent
calculations of the nuclear matrix elements (NMEs), we show that while the
claim in $^{76}$Ge is still compatible with the individual limits from
$^{136}$Xe, it is inconsistent with the KamLAND-Zen+EXO-200 combined limit for
all but one NME calculations. After imposing the most stringent upper limit on
the sum of light neutrino masses from Planck, we find that the canonical light
neutrino contribution cannot satisfy the claimed $0\nu\beta\beta$ signature or
saturate the current limit, irrespective of the NME uncertainties. However,
inclusion of the heavy neutrino contributions, arising naturally in TeV-scale
Left-Right symmetric models, can saturate the current limit of
$0\nu\beta\beta$. In a type-II seesaw framework, this imposes a lower limit on
the lightest neutrino mass. Depending on the mass hierarchy, we obtain this
limit to be in the range of 0.07 - 4 meV for a typical choice of the
right-handed (RH) gauge boson and RH neutrino masses relevant for their
collider searches. Using the $0\nu\beta\beta$ bounds, we also derive correlated
constraints in the RH sector, complimentary to those from the LHC.
