Distribution grid topology and admittance information are essential for system planning, operation, and protection. In many distribution grids, missing or inaccurate topology and admittance data call for efficient estimation methods. However, measurement data may be insufficient or contaminated with large noise, which will fundamentally limit the estimation accuracy. This work explores the theoretical precision limits of the topology and admittance estimation (TAE) problem with different measurement devices, noise levels, and numbers of measurements. On this basis, we propose a conservative progressive self-adaptive (CPS) algorithm to estimate the topology and admittance. The results on IEEE 33 and 141-bus systems validate that the proposed CPS method can approach the theoretical precision limits under various measurement settings.