There is a wide consensus that Type Ia supernovae (SNe Ia) originate from the thermonuclear explosion of CO white dwarfs (WDs), with the lack of hydrogen in the observed spectra as a distinctive feature. Here, we present supernova (SN) 2016jae, which was classified as an SN Ia from a spectrum obtained soon after its discovery. The SN reached a B-band peak of -17.93 +/- 0.34 mag, followed by a fast luminosity decline with s(BV)0.56 +/- 0.06 and inferred Delta m(15)(B) of 1.88 +/- 0.10 mag. Overall, the SN appears to be a 'transitional' event between a 'normal' SN Ia and a very dim SN Ia, such as 91bg-like SNe. Its peculiarity is that two late-time spectra, taken at +84 and +142 days after the peak, show a narrow line of H alpha (with full width at half maximum of similar to 650 and 1000 km s(-1), respectively). This is the third low-luminosity and fast-declining SN Ia, after SN2018cqj/ATLAS18qtd and SN2018fhw/ASASSN-18tb, found in the 100IAS survey to show a resolved narrow H alpha line in emission in its nebular-phase spectra. We argue that the nebular H alpha emission originates in an expanding hydrogen-rich shell (with velocity <= 1000 km s(-1)). The hydrogen shell velocity is too high to be produced during a common envelope phase, though it may be consistent with some material stripped from an H-rich companion star in a single-degenerate progenitor system. However, the derived mass of this stripped hydrogen is similar to 0.002-0.003 M-circle dot, which is much less than that expected (> 0.1 M-circle dot) from standard models for these scenarios. Another plausible sequence of events is a weak SN ejecta interaction with an H shell ejected by optically thick winds or a nova-like eruption on the CO WD progenitor some years before the SN explosion.