We explore the origin of mid-infrared (mid-IR) dust extinction in all 20 nearby (z < 0.05) bona-fide Compton-thick (N_H > 1.5 x 10^24 cm^-2) AGN with hard energy (E > 10 keV) X-ray spectral measurements. We accurately measure the silicate absorption features at lambda~9.7um in archival low-resolution (R~57-127) Spitzer Infrared Spectrograph (IRS) spectroscopy, and show that only a minority (~45%) of nearby Compton-thick AGN have strong Si-absorption features (S_9.7 = ln(f_{int}/f_{obs}) > 0.5) which would indicate significant dust attenuation. The majority (~60%) are star-formation dominated (AGN:SB<0.5) at mid-IR wavelengths and lack the spectral signatures of AGN activity at optical wavelengths, most likely because the AGN emission-lines are optically-extinguished. Those Compton-thick AGN hosted in low-inclination angle galaxies exhibit a narrow-range in Si-absorption (S_9.7 ~ 0-0.3), which is consistent with that predicted by clumpy-torus models. However, on the basis of the IR spectra and additional lines of evidence, we conclude that the dominant contribution to the observed mid-IR dust extinction is dust located in the host galaxy (i.e., due to disturbed morphologies; dust-lanes; galaxy inclination angles) and not necessarily a compact obscuring torus surrounding the central engine.