It is a common practice to use ideal thermal boundary conditions to investigate natural convection. These correspond to very good conducting walls and to very bad conducting walls. In particular, this has been the case in natural convection of viscoelastic fluids. In this paper, these conditions are generalized by taking into account the finite thermal conductivities and thicknesses of the walls in the natural convection of a viscoelastic Jeffreys fluid heated from below. The goal is to present more realistic results related to experimental conditions. The critical Rayleigh number R_{c}, the frequency of oscillation ω_{c}, and the wavenumber k_{c} have been plotted varying the properties of the walls from the case of very good thermal conductivity to very poor thermal conductivity. In order to understand the convective phenomena, two parameters are fixed and the other one varied among the nondimensional relaxation time F, the relative retardation time E, and the Prandtl number Pr of the viscoelastic fluid. The role of the relative retardation time E on the thermal instability is discussed in detail.