  Establishing if multi-Higgs potentials are bounded from below (BFB) can be
rather challenging, and it may impede efficient investigation of all
phenomenological consequences of such models. In this paper, we find the
necessary and sufficient BFB conditions for the Three-Higgs-Doublet model
(3HDM) with the global symmetry group $U(1)\times U(1)$. We observed an
important role played by charge-breaking directions in the Higgs space, even
for situations when a good-looking neutral minimum exists. This remark is not
limited to the particular model we consider but represents a rather general
feature of elaborate multi-Higgs potentials which must be carefully dealt with.
Also, applying this method to Weinberg's model (the $\mathbb{Z}_2 \times
\mathbb{Z}_2$ symmetric 3HDM) turned out to be more challenging than was
believed in the literature. In particular, we have found that the approach
taken in a paper from 2009 does not lead to the necessary and sufficient BFB
conditions for this case.
