To engineer tunable thin film materials, accurate measurement of their mechanical properties is crucial. However, characterizing the elastic modulus with current methods is particularly challenging for sub-micrometer thick films and hygroscopic materials because they are highly sensitive to environmental conditions and most methods require free-standing films which are difficult to prepare. In this work, we directly compared three buckling-based methods to determine the elastic moduli of supported thin films: 1) biaxial thermal shrinking, 2) uniaxial thermal shrinking, and 3) the mechanically compressed, strain-induced elastic buckling instability for mechanical measurements (SIEBIMM) method. Nanobiocomposite model films composed of cellulose nanocrystals (CNCs) and polyethyleneimine (PEI) were assembled using layer-by-layer deposition to control composition and thickness. The three buckling-based methods yielded the same trends and comparable values for the elastic moduli of each CNC-PEI film composition (ranging from 15 – 44 GPa, depending on film composition). This suggests that the methods are similarly effective for the quantification of thin film mechanical properties. Increasing the CNC content in the films statistically increased the modulus, however, increasing the PEI content did not lead to significant changes. The standard deviation of elastic moduli determined from SIEBIMM was 2-4 times larger than for thermal shrinking, likely due to extensive cracking and partial film delamination. In light of these results, biaxial thermal shrinking is recommended as the method of choice because it affords the simplest implementation and analysis and is the least sensitive to small deviations in the input parameter values, such as film thickness or substrate modulus.