Left-over, ablated material from a possible non-degenerate companion can reveal itself after about one year in spectra of Type Ia SNe (SNe Ia). We have searched for such material in spectra of SN 2011fe (at 294 days after the explosion) and for SN 2014J (315 days past explosion). The observations are compared with numerical models simulating the expected line emission. The spectral lines sought for are H-alpha, [O I] 6300 and [Ca II] 7291,7324, and the expected width of these lines is about 1000 km/s. No signs of these lines can be traced in any of the two supernovae. When systematic uncertainties are included, the limits on hydrogen-rich ablated gas in SNe 2011fe and 2014J are 0.003 M_sun and 0.0085 M_sun, respectively, where the limit for SN 2014J is the second lowest ever, and the limit for SN 2011fe is a revision of a previous limit. Limits are also put on helium-rich ablated gas. These limits are used, in conjunction with other data, to argue that these supernovae can stem from double-degenerate systems, or from single-degenerate systems with a spun up/spun down super-Chandrasekhar white dwarf. For SN 2011fe, other types of hydrogen-rich donors can likely be ruled out, whereas for SN 2014J a main-sequence donor system with large intrinsic separation is still possible. Helium-rich donor systems cannot be ruled out for any of the two supernovae, but the expected short delay time for such progenitors makes this possibility less likely, especially for SN 2011fe. The broad [Ni II] 7378 emission in SN 2014J is redshifted by about +1300 km/s, as opposed to the known blueshift of roughly -1100 km/s for SN 2011fe. [Fe II] 7155 is also redshifted in SN 2014J. SN 2014J belongs to a minority of SNe Ia that both have a nebular redshift of [Fe II] 7155 and [Ni II] 7378, and a slow decline of the Si II 6355 absorption trough just after B-band maximum.
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