  The constraints from the measurements of the $B\to X_s\gamma$ decay rate on
the parameter space of 3-Higgs Doublet Models (3HDMs), where all the doublets
have non-zero vacuum expectation values, are studied at the next-to-leading
order in QCD. In order to naturally avoid the presence of flavour changing
neutral currents at the tree level, we impose two softly-broken discrete $Z_2$
symmetries. This gives rise to five independent types of 3HDMs that differ in
their Yukawa couplings. We show that in all these 3HDMs (including the case of
type-II-like Yukawa interactions) both masses of the two charged Higgs bosons
$m_{H_1^\pm}$ and $m_{H_2^\pm}$ can be smaller than the top mass $m_t$ while
complying with the constraints from $B\to X_s\gamma$. As an interesting
phenomenological consequence, the branching ratios of the charged Higgs bosons
decay into the $cb$ final states can be as large as $80\%$ when their masses
are taken to be below $m_t$ in two of the five 3HDMs (named as Type-Y and
Type-Z). This light charged Higgs boson scenario provides a hallmark 3HDM
signature that cannot be realised in $Z_2$ symmetric 2-Higgs doublet models. We
find that in the Type-Y and Type-Z 3HDMs the scenario with $90\,{\rm GeV}<
m_{H^\pm_1}^{}\,, m_{H^\pm_2}^{}<m_t$ is ruled out by the direct searches at
the LHC, but in the Type-Y 3HDM $80~\text{GeV}<m_{H_1^\pm}^{}<90~\text{GeV}$
and $90~\text{GeV}<m_{H_2^\pm}^{}<m_t$ is allowed by $B\to X_s\gamma$ and
direct searches at LEP2, Tevatron and LHC due to the reduced sensitivity of
these searches to the degenerate case $m_{H_1^\pm}\approx m_{W^\pm}$. The cases
where only one or both charged Higgs bosons are above the top quark mass are
also naturally allowed in the both Type-Y and Type-Z 3HDMs.
