We present deep Chandra X-ray observations of two nearby Type Ia supernovae, SN 2017cbv and SN 2020nlb, which reveal no X-ray emission down to a luminosity L_X_<~5.3x10^37^ and <~5.4x10^37^erg/s (0.3-10keV), respectively, at ~16-18 days after the explosion. With these limits, we constrain the pre-explosion mass-loss rate of the progenitor system to be dM/dt<7.2x10^-9^ and <9.7x10^-9^M_{sun}_/yr for each (at a wind velocity v_w_=100km/s and a radius of R~10^16^cm), assuming any X-ray emission would originate from inverse Compton emission from optical photons upscattered by the supernova shock. If the supernova environment was a constant-density medium, we would find a number density limit of n_CSM_<36 and <65cm^-3^, respectively. These X-ray limits rule out all plausible symbiotic progenitor systems, as well as large swathes of parameter space associated with the single degenerate scenario, such as mass loss at the outer Lagrange point and accretion winds. We also present late-time optical spectroscopy of SN2020nlb, and set strong limits on any swept up hydrogen (L_H{alpha}_<2.7x10^37^erg/s) and helium (L_He,{lambda}6678_<2.7x10^37^erg/s) from a nondegenerate companion, corresponding to M_H_<~0.7-2x10^-3^M_{sun}_ and M_He_<~4x10^-3^M_{sun}_. Radio observations of SN 2020nlb at 14.6 days after explosion also yield a non-detection, ruling out most plausible symbiotic progenitor systems. While we have doubled the sample of normal Type Ia supernovae with deep X-ray limits, more observations are needed to sample the full range of luminosities and subtypes of these explosions, and set statistical constraints on their circumbinary environments.