  In supersymmetric $E_6$ the masses of the third family quarks and charged
lepton, $t-b-\tau$, as well as the masses of the vector-like quarks and
leptons, $D-\bar{D}$ and $ L-\bar{L}$, may arise from the coupling $27_3$ x
$27_3$ x $27_H$, where $27_3$ and $27_H$ denote the third family matter and
Higgs multiplets respectively. We assume that the SO(10) singlet component in
$27_H$ acquires a TeV scale VEV which spontaneously breaks U(1)$_\psi$ and
provides masses to the vector-like particles in $27_3$, while the MSSM doublets
in $27_H$ provide masses to $ t, b$ and $\tau$. Imposing Yukawa coupling
unification $h_t=h_b=h_{\tau}=h_D=h_L$ at $M_{GUT}$ and employing the ATLAS and
CMS constraints on the $Z'_\psi$ boson mass, we estimate the lower bounds on
the third family vector-like particles $D-\bar{D}$ and $L-\bar{L}$ masses to be
around 5.85 TeV and 2.9 TeV respectively. These bounds apply in the
supersymmetric limit.
