We analyze the sensitivity of the flame propagation in a Chandrasekhar mass white dwarf (WD) to initial conditions during the subsonic burning phase (deflagration) using two-dimensional simulations of the full WD. Results are presented for a wide variety of initial flame distributions, including central and off-center, single-point and multipoint, and simultaneous and nonsimultaneous ignitions. We also examine the effects of the convective velocity field that should exist at the core before thermonuclear runaway. Our main conclusion suggests that the amounts of burning products and their distributions through the deflagration phase are extremely sensitive to initial conditions, much more sensitive than those presented in previous studies. In particular, we find that more complex configurations, such as even slightly off-center ignitions, nonsimultaneous multipoint ignitions, and velocity fields, tend to favor solutions in which individual plumes rise faster than the bulk of a typical Rayleigh-Taylor-driven, unstable burning front. The difference from previous calculations for an octant of a WD may be understood as a consequence of the suppression of the l = 1 and 2 modes. Our results are consistent with full-star three-dimensional calculations by the Chicago group. Moreover, the total amount of nuclear burning during the phase of subsonic burning depends sensitively on the initial conditions and may cause the WD to become unbound or to pulsate. The latter case opens (again) a new parameter space for the Chandrasekhar WD problem. We discuss the implications of the results on current models for Type Ia supernovae and limitations imposed by the two-dimensional nature of our study and suggest directions for further study.