We study the entanglement of families of Unruh modes in the Bell states |Φ±⟩=1/2√(|00⟩±|11⟩) and |Ψ±⟩=1/2√(|01⟩±|10⟩) shared by two accelerated observers and find fundamental differences in the robustness of entanglement against acceleration for these states. States Ψ± are entangled for all finite accelerations, whereas, due to the Unruh effect, states Φ± lose their entanglement for finite accelerations. This is true for Bell states of two bosonic modes, as well as for Bell states of a bosonic and a fermionic mode. Furthermore, there are also differences in the degradation of entanglement for Bell states of fermionic modes. We reveal the origin of these distinct characteristics of entanglement degradation and discuss the role that is played by particle statistics. Our studies suggest that the behavior of entanglement in accelerated frames strongly depends on the occupation patterns of the constituent states, whose superposition constitutes the entangled state, where especially states Φ± and Ψ± exhibit distinct characteristics regarding entanglement degradation. Finally, we point out possible implications of hovering over a black hole for these states.