We combine the equation of state of dense matter up to twice nuclear saturation density nsat obtained using chiral effective field theory (χ EFT ) and recent observations of neutron stars to gain insights about the high-density matter encountered in their cores. A key element in our study is the recent Bayesian analysis of correlated EFT truncation errors based on order-by-order calculations up to next-to-next-to-next-to-leading order in the χ EFT expansion. We refine the bounds on the maximum mass imposed by causality at high densities and provide stringent limits on the maximum and minimum radii of ∼1.4 M⊙ and ∼2.0 M⊙ stars. Including χ EFT predictions from nsat to 2 nsat reduces the permitted ranges of the radius of a 1.4 M⊙ star, R1.4, by ∼3.5 km . If observations indicate R1.4<11.2 km , then our study implies that either the squared speed of sound cs2>1 /2 for densities above 2 nsat or that χ EFT breaks down below 2 nsat . We also comment on the nature of the secondary compact object in GW190814 with mass ≃2.6 M⊙ and discuss the implications of massive neutron stars >2.1 M⊙(2.6 M⊙) in future radio and gravitational-wave searches. Some form of strongly interacting matter with cs2>0.35 (0.55 ) must be realized in the cores of such massive neutron stars. In the absence of phase transitions below 2 nsat , the small tidal deformability inferred from GW170817 lends support for the relatively small pressure predicted by χ EFT for the baryon density nB in the range 1 -2 nsat . Together they imply that the rapid stiffening required to support a high maximum mass should occur only when nB≳1.5 -1.8 nsat .