Abstract Li₃((LiCr)(Te/Sb))O₆ compounds where Cr atoms along with Li and Te or Sb are part of a honeycomb and are studied using magnetic susceptibility, specific heat, x-ray photoelectron spectroscopy and neutron diffraction. The oxides stoichiometries as determined from the neutron diffraction studies are Li_4.47Cr_0.53TeO₆ and Li_3.88Cr_1.12SbO₆ with a stable oxidation state of +3 for Cr. Both the compounds crystallize in space group C2/m with intermixing of cations at the 4g sites leaving the 2a sites preferentially for Te or Sb. Again, the Li⁺ ions alone predominantly occur in the interlayer sites. Both the compounds show a broad anomaly in specific heat at 8 K, which is robust against 8 T. A corresponding anomaly is absent in the magnetic susceptibility but recovers from its derivative, dχ(T)/dT. We ascertain the magnetic anomaly temperatures (T _a ) of Li_4.47Cr_0.53TeO₆ and Li_3.88Cr_1.12SbO₆ as 5.9 K and 6.7 K respectively from specific heat. Although the physical properties indicated a low temperature anomaly, neutron diffraction data did not reveal a magnetic signal or a structural anomaly down to 1.5 K. This rules out a conventional long-range ordered magnetic ground state in either compounds. Combining the results from specific heat, neutron diffraction and electron paramagnetic resonance, we put forth a scenario of depleted honeycomb lattice of Cr³⁺ with predominant short-range magnetic correlations as the magnetic ground states of the title compounds.