Abstract : We present optical photometry and spectroscopy of five type Ia supernovae discovered by the Nearby Supernova Factory selected to be spectroscopic analogues of the candidate super-Chandrasekhar-mass events SN 2003fg and SN 2007if. Their spectra are characterized by hot, highly ionized photospheres near maximum light, for which SN 1991T supplies the best phase coverage among available close spectral templates. Like SN 2007if, these supernovae are overluminous (-19.5 < M_V < -20) and the velocity of the Si II 6355 absorption minimum is consistent with being constant in time from phases as early as a week before, and up to two weeks after, $B$-band maximum light. We interpret the velocity plateaus as evidence for a reverse-shock shell in the ejecta formed by interaction at early times with a compact envelope of surrounding material, as might be expected for SNe resulting from the mergers of two white dwarfs. We use the bolometric light curves and line velocity evolution of these SNe to estimate important parameters of the progenitor systems, including nickel-56 mass, total progenitor mass, and masses of shells and surrounding carbon/oxygen envelopes. We find that the reconstructed total progenitor mass distribution of the events (including SN 2007if) is bounded from below by the Chandrasekhar mass, with SN 2007if being the most massive. We discuss the relationship of these events to the emerging class of super-Chandrasekhar-mass SNe Ia, estimate the relative rates, compare the mass distribution to that expected for double-degenerate SN Ia progenitors from population synthesis, and consider implications for future cosmological Hubble diagrams.