  We present electrical resistivity and ac-susceptibility measurements of
GdTe$_3$, TbTe$_3$ and DyTe$_3$ performed under pressure. An upper
charge-density-wave (CDW) is suppressed at a rate of
$\mathrm{d}T_{\mathrm{CDW,1}}/\mathrm{d}P$ = $-$85 K/GPa. For TbTe$_3$ and
DyTe$_3$, a second CDW below $T_{\mathrm{CDW,2}}$ increases with pressure until
it reaches the $T_{\mathrm{CDW,1}}$($P$) line. For GdTe$_3$, the lower CDW
emerges as pressure is increased above $\sim$ 1 GPa. As these two CDW states
are suppressed with pressure, superconductivity (SC) appears in the three
compounds at lower temperatures. Ac-susceptibility experiments performed on
TbTe$_3$ provide compelling evidence for bulk SC in the low-pressure region of
the phase diagram. We provide measurements of superconducting critical fields
and discuss the origin of a high-pressure superconducting phase occurring above
5 GPa.
