A current goal driving alloy development is the identification of alloy compositions for high temperature applications but with the additional requirement of sufficient ductility at ambient temperatures. Multicomponent, single-phase, polycrystalline High Entropy Alloys (HEAs) have recently emerged as a new class of metal alloys, and some refractory bcc HEAs composed mainly of Nb, V, Ta, Cr, Mo, and/or W show excellent strength retention up to very high temperatures but low ductility at room temperature (RT). Here, it is postulated that the macroscopic ductility in bcc elements and alloys is determined by the intrinsic competition between brittle cleavage and ductile dislocation emission mechanisms at an atomistically sharp crack. The stress intensities K_Ic for cleavage and K_Ie for emission are evaluated within Linear Elastic Fracture Mechanics and validated by atomistic simulations on model alloys. A RT ductility criterion based on K_Ie/K_Ic for critical crack orientations is proposed based on the elemental metals and is then applied to HEAs. Agreement with experimental trends in ductility vs. composition across a range of existing HEAs is demonstrated. The analysis is then extended across large composition spaces of the Mo-Nb-Ta-V-W and Mo-Nb-Ti alloy families, identifying new compositions with the potential for RT ductility.