  Motivated by the peculiar features observed through intrinsic tunneling
spectroscopy of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ mesas in the normal state,
we have extended the normal state two-barrier model for the c-axis transport
[M. Giura et al., Phys. Rev. B {\bf 68}, 134505 (2003)] to the analysis of
$dI/dV$ curves. We have found that the purely normal-state model reproduces all
the following experimental features: (a) the parabolic $V$-dependence of
$dI/dV$ in the high-$T$ region (above the conventional pseudogap temperature),
(b) the emergence and the nearly voltage-independent position of the "humps"
from this parabolic behavior lowering the temperature, and (c) the crossing of
the absolute $dI/dV$ curves at a characteristic voltage $V^\times$. Our
findings indicate that conventional tunneling can be at the origin of most of
the uncommon features of the c axis transport in
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$. We have compared our calculations to
experimental data taken in severely underdoped and slightly underdoped
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ small mesas. We have found good agreement
between the data and the calculations, without any shift of the calculated
dI/dV on the vertical scale. In particular, in the normal state (above
$T^\ast$) simple tunneling reproduces the experimental dI/dV quantitatively.
Below $T^\ast$ quantitative discrepancies are limited to a simple rescaling of
the voltage in the theoretical curves by a factor $\sim$2. The need for such
modifications remains an open question, that might be connected to a change of
the charge of a fraction of the carriers across the pseudogap opening.
