In this article we conduct an analytical study of a poroviscoelastic mixture model stemming from the classical Biot's consolidation model for poroelastic media, comprising a fluid component and a solid component, coupled with a viscoelastic stress-strain relationship for the total stress tensor. The poroviscoelastic mixture is studied in the one-dimensional case, corresponding to the experimental conditions of confined compression. Upon assuming (i) negligible inertial effects in the balance of linear momentum for the mixture, (ii) a Kelvin-Voigt model for the effective stress tensor and (iii) a constant hydraulic permeability, we obtain an initial value/boundary value problem of pseudo-parabolic type for the spatial displacement of the solid component of the mixture. The dimensionless form of the differential equation is characterized by the presence of two positive parameters γ and η, representing the contributions of compressibility and structural viscoelasticity, respectively. Explicit solutions are obtained for different functional forms characterizing the boundary traction. The main result of our analysis is that the compressibility of the components of a poroviscoelastic mixture does not give rise to unbounded responses to non-smooth traction data. Interestingly, compressibility allows the system to store potential energy as its components are elastically compressed, thereby providing an additional mechanism that limits the maximum of the discharge velocity when the imposed boundary traction is irregular in time.