The light curves of Type Ia supernovae (SNe Ia) are powered by the radioactive decay of ^56^Ni to ^56^Co at early times, and the decay of ^56^Co to ^56^Fe from ~60 d after explosion. We examine the evolution of the [CoIII] {lambda}5893 emission complex during the nebular phase for SNe Ia with multiple nebular spectra and show that the line flux follows the square of the mass of ^56^Co as a function of time. This result indicates both efficient local energy deposition from positrons produced in ^56^Co decay and long-term stability of the ionization state of the nebula. We compile SN Ia nebular spectra from the literature and present 21 new late-phase spectra of 7 SNe Ia, including SN 2014J. From these we measure the flux in the [CoIII] {lambda}5893 line and remove its well-behaved time dependence to infer the initial mass of ^56^Ni (M_Ni_) produced in the explosion. We then examine ^56^Ni yields for different SN Ia ejected masses (M_ej_ - calculated using the relation between light-curve width and ejected mass) and find that the ^56^Ni masses of SNe Ia fall into two regimes: for narrow light curves (low stretch s~0.7-0.9), M_Ni_ is clustered near M_Ni_~0.4M_{sun}_ and shows a shallow increase as M_ej_ increases from ~1 to 1.4M_{sun}_; at high stretch, M_ej_ clusters at the Chandrasekhar mass (1.4M_{sun}_) while M_Ni_ spans a broad range from 0.6 to 1.2M_{sun}_. This could constitute evidence for two distinct SN Ia explosion mechanisms.