Quick answer
They can, but the likelihood is lower than on bare conductor. On a bare line, a bird or monkey bridging two phases, or a phase and an earthed part, can start a fault immediately. A covered conductor places an insulating layer between the animal and the metal, so brief contact is less likely to cause a trip.
The covering is not a guarantee. Gaps at poles, bare fittings and damaged sections can still be points of contact. The APAR covered conductor product page lists forests, dense vegetation and wildlife sanctuaries among its intended applications. Fit matched accessories and insulate any exposed hardware to get the full benefit.