  In several searches for additional Higgs bosons at the LHC, in particular in
a CMS search exploring decays to pairs of top quarks, $t \bar t$, and in an
ATLAS search studying tau leptons, $\tau^+\tau^-$, local excesses of about
$3\,\sigma$ standard deviations or above have been observed at a mass scale of
approximately $ 400$GeV. We investigate to what extent a possible signal in
these channels could be accommodated in the Next-to-Two-Higgs-Doublet Model
(N2HDM) or the Next-to Minimal Supersymmetric Standard Model (NMSSM). In a
second step we analyze whether such a model could be compatible with both a
signal at around $400$GeV and $96$GeV, where the latter possibility is
motivated by observed excesses in searches for the $b \bar b$ final state at
LEP and the di-photon final state at CMS. The analysis for the N2HDM reveals
that the observed excesses at $400GeV$ in the $t \bar t$ and $\tau^+\tau^-$
channels point towards different regions of the parameter space, while one such
excess and an additional Higgs boson at around $96$GeV could simultaneously be
accommodated. In the context of the NMSSM an experimental confirmation of a
signal in the $t \bar t$ final state would favour the
alignment-without-decoupling limit of the model, where the Higgs boson at
$125$GeV could be essentially indistinguishable from the Higgs boson of the
standard model. In contrast, a signal in the $\tau^+\tau^-$ channel can only be
accommodated outside of this limit, and parts of the investigated parameter
space could be probed with Higgs signal-rate measurements at the (HL-)LHC.
