In this work, we show that the hybrid material Co2(1,4-bdc)2(dabco)·[4DMF·1H2O], shows an apparent colossal dielectric constant at room temperature (ε′r ≈ 5000 at 300 K for ν = 100 Hz). Nevertheless, such response does not imply colossal polarizability processes, as its dielectric constant is not purely intrinsic, but is greatly enhanced by the activation of extrinsic dielectric effects close to room temperature associated to the diffusion of numerous guest molecules through the channels. If such extrinsic contributions are eliminated or reduced, the values of the dielectric constant turn to be much smaller, as observed in the closely related Co2(1,4-bdc-NH2)2(dabco)·[7/2DMF·1H2O], Co2(1,4-ndc)2(dabco) ·[3DMF·2H2O] and Ni2(1,4-bdc)2(dabco)·[3DMF·1/2H2O] compounds. Therefore, we warn about the imperious necessity of distinguishing between intrinsic and extrinsic effects in electrically inhomogenous MOF materials that display a certain conductivity in order to adequately interpret their dielectric behavior.