  Recent transport experiments have revealed the activation of longitudinal
magnetoresistance of Weyl semimetals in the quantum limit, suggesting the
breakdown of chiral anomaly in a strong magnetic field. Here we provide a
general mechanism for gapping the zeroth chiral Landau levels applicable for
both Dirac and Weyl semimetals. Our result shows that the zeroth Landau levels
anticross when the magnetic axis is perpendicular to the Dirac/Weyl node
separation and when the inverse magnetic length $l_B^{-1}$ is comparable to the
node separation scale $\Delta k$. The induced bulk gap increases rapidly beyond
a threshold field in Weyl semimetals, but has no threshold and is non-monotonic
in Dirac systems due to the crossover between $l_B^{-1}>\Delta k$ and
$l_B^{-1}<\Delta k$ regions. We also find that the Dirac and possibly Weyl
systems host counterpropagating edge states between the zeroth Landau levels,
leading to a state with metallic side walls and zero Hall conductance.
