X-ray polarimetry is largely an unexplored area of an otherwise mature field of X-ray astronomy. Except for a few early attempts during the 1970s, no dedicated X-ray polarimeter has been flown during the past four decades. On the other hand, the scientific value of X-ray polarization measurement has been well known for a long time, and there has been significant technical progress in developing sensitive X-ray polarimeters in recent years. But there are no approved dedicated X-ray polarimetric experiments to be flown in the near future, so it is important to explore the polarimetric capabilities of other existing or planned instruments and examine whether they can provide significant astrophysical polarization measurements. In this paper, we present experimental results to show that the CZTI instrument onboard the forthcoming Indian astronomy mission, Astrosat, will be able to provide sensitive measurements of X-ray polarization in the energy range of 100-300 keV. CZTI will be able to constrain any intrinsic polarization greater than ~40% for bright X-ray sources (>500 mCrab) within a short exposure of ~100 ks with a 3-sigma confidence level. We show that this seemingly "modest" sensitivity can play a very significant role in addressing long pending questions, such as the contribution of relativistic jets to hard X-rays in black hole binaries and X-ray emission mechanism and geometry in X-ray pulsars.