We study fracture propagation in bi-layered materials using an adaptive phase-field method. The method combines a phase-field representation of fracture with spatial adaptivity to provide mesh refinement in the smeared zone of damage, where higher accuracy is needed to resolve the solution. We place a particular emphasis on studying the competition between crack arrest, deflection, and penetration at the interface of two dissimilar materials. To explore this competition, we perform tension tests on various models with different combinations of material and geometric properties. We begin by validating the adaptive phase-field model through a uniaxial tension test. The material mismatch and locations of the initial damage for this example are chosen such that the fracture propagation behavior is identical to that in a homogeneous material. Next, we independently vary the mismatches in elastic stiffness and critical energy release rate to study their influence on crack growth. Our observations show that both mismatches contribute to an effective toughening of the model, but in different ways. Critical energy release rate mismatch causes crack deflection along the interfaces, while elastic stiffness mismatch results in retardation of crack growth. We also note that for a specific combination of material mismatch and location of initial damage, crack nucleation in the adjacent material is preferred over propagation from the existing notch. Finally, through models with inclined interfaces, we demonstrate the effect of the relative orientation between the approaching crack and the material interface on crack penetration, deflection, and nucleation. The numerical results from this study provide valuable insight into the various failure mechanisms in bi-layered materials.