We place statistical constraints on Type Ia supernova (SN Ia) progenitors using 227 nebular-phase spectra of 111 SNe Ia. We find no evidence of stripped companion emission in any of the nebular-phase spectra. Upper limits are placed on the amount of mass that could go undetected in each spectrum using recent hydrodynamic simulations. With these null detections, we place an observational 3{sigma} upper limit on the fraction of SNe Ia that are produced through the classical H-rich non-degenerate companion scenario of <5.5 per cent. Additionally, we set a tentative 3{sigma} upper limit on He star progenitor scenarios of <6.4 per cent, although further theoretical modelling is required. These limits refer to our most representative sample including normal, 91bg-like, 91T-like, and 'super-Chandrasekhar' SNe Ia but excluding SNe Iax and SNe Ia-CSM. As part of our analysis, we also derive a Nebular Phase Phillips Relation, which approximates the brightness of an SN Ia from 150 to 500d after maximum using the peak magnitude and decline rate parameter {Delta}m_15_(B).