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, $T_{1/2}^{2 uvarepsilonbeta^+}geq 1.1times 10^{21}$ yr, has reached the region of theoretical predictions. With this half-life limit the effective nuclear matrix element for the $2 uvarepsilonbeta^+$ decay is bounded as $M^{2 uvarepsilonbeta^+}_{eff}le 1.1$. 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 $T_{1/2} geq (8.5times10^{20}-1.4times10^{21}$) yr.
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