As highlighted in the image below, for cable-supported bridges, because of the deck's lower stiffness and cables' additional flexibility, it's the vertical deck displacements and not the tower longitudinal displacements that are usually limited by the vertical live loads. 

Moreover, it's usual that the displacement worst-case scenarios might appear in the construction stage analysis: Probably in the final erection stages, when the structure is still "open" and the last segment is under construction, yet in a balanced cantilever method (BCM).

Once closed, the structural model changes to a more rigid frame and consequentially less flexible. The mast's structure stress, crack, or fatigue are then the subject of careful study as limitations. Some structures might have "floating" decks not totally connected to the masts.

However, for horizontal longitudinal and lateral forces, Wind and Seismic are generally the limit for cable-supported bridges, not the traffic or accident loading. Also, it's more of a vibration issue and not displacement only. Therefore, when needed, control systems to dissipate the input energy are widely used. For example, the passive control, semi-active control, or active control systems. Among the typical passive control systems, the viscous damper (VD) and elastic connecting devices (ECDs) are widely used with emphasis on Tuned Mass Dampers (TMD).