  A radiopure cadmium tungstate crystal scintillator, enriched in $^{106}$Cd to
66%, with mass of 216 g ($^{106}$CdWO$_4$), was used to search for double beta
decay processes in $^{106}$Cd in coincidence with four ultra-low background
high purity germanium detectors in a single cryostat. New improved limits on
the double beta processes in $^{106}$Cd have been set on the level of $10^{20}-
10^{21}$ yr after 13085 h of data taking. In particular, the half-life limit on
the two neutrino electron capture with positron emission,
$T_{1/2}^{2\nu\varepsilon\beta^+}\geq 1.1\times 10^{21}$ yr, has reached the
region of theoretical predictions. With this half-life limit the effective
nuclear matrix element for the $2\nu\varepsilon\beta^+$ decay is bounded as
$M^{2\nu\varepsilon\beta^+}_{eff}\le 1.1$. The resonant neutrinoless double
electron captures to the 2718 keV, 2741 keV and 2748 keV excited states of
$^{106}$Pd are restricted at the level of $T_{1/2} \geq
(8.5\times10^{20}-1.4\times10^{21}$) yr.
