Both Nb$_3$Pd$_x$Se$_7$ and Ta$_4$Pd$_3$Te$_{16}$ crystallize in a monoclinic point group while exhibiting superconducting transition temperatures as high as $T_csim 3.5$ and $sim 4.7 $ K, respectively. Disorder was claimed to lead to the extremely large upper critical fields ($H_{c2}$) observed in related compounds. Despite the presence of disorder and heavier elements, $H_{c2}$s in Ta$_4$Pd$_3$Te$_{16}$ are found to be considerably smaller than those of Nb$_3$Pd$_x$Se$_7$ while displaying an anomalous, non-saturating linear dependence on temperature $T$ for fields along all three crystallographic axes. In contrast, crystals of the latter compound displaying the highest $T_c$s display $H_{c2}propto (1-T/T_c)^{1/2}$, which in monolayers of transition metal dichalcogenides is claimed to be evidence for an Ising paired superconducting state resulting from strong spin-orbit coupling. This anomalous $T$-dependence indicates that the superconducting state of Nb$_3$Pd$_x$Se$_7$ is quasi-two-dimensional in nature. This is further supported by a nearly divergent anisotropy in upper-critical fields, i.e. $gamma= H_{c2}^{b}/H_{c2}^{a^{prime}}$, upon approaching $T_c$. Hence, in Nb$_3$Pd$_x$Se$_7$ the increase of $T_c$ correlates with a marked reduction in electronic dimensionality as observed, for example, in intercalated FeSe. For the Nb compound, Density functional theory (DFT) calculations indicate that an increase in the external field produces an anisotropic orbital response, with especially strong polarization at the Pd sites when the field is perpendicular to their square planar environment. Therefore, DFT suggests the field-induced pinning of the spin to the lattice as a possible mechanism for decoupling the superconducting planes. Overall, our observations represent further evidence for unconventional superconductivity in the Pd chalcogenides.
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