This study explores the flexural behavior of continuous fiber-reinforced composite sandwich structures built entirely using material extrusion additive manufacturing. The continuous fiber additive manufacturing system used in this study works sequentially, thus enabling the addition of fiber reinforcement just in the face sheets, where it is most effective. Three-point bending tests were carried out on sandwich panel specimens built using thermoplastic reinforced with continuous glass fiber to quantify the effect of fiber reinforcement and infill density in the flexural properties and failure mode. Sandwich structures containing continuous fiber reinforcement had higher flexural strength and rigidity than unreinforced sandwiches. On the other hand, an increase in the lattice core density did not improve the flexural strength and rigidity. The elastic modulus of fiber-reinforced 3D-printed sandwich panels exceeded the predictions of the analytical models; the equivalent homogeneous model had the best performance, with a 15% relative error. However, analytical models could not correctly predict the failure mode: wrinkle failure occurs at 75% and 30% of the critical load in fiber-reinforced sandwiches with low- and high-density cores, respectively. Furthermore, no model is currently available to predict interlayer debonding between the matrix and the thermoplastic coating of fiber layers. Divergences between analytical models and experimental results could be attributed to the simplifications in the models that do not consider defects inherent to additive manufacturing, such as air gaps and poor interlaminar bonding.