Nonlinear hydrodynamic stability analysis has been done for viscoelastic fluids heated from below for the cases of rigid-rigid and ridgid-free boundary conditions that can be compared with experimental results. We adopted a very general constitutive relation which encompasses the Maxwell model, the Jeffreys model and the Phan-Thien-Tanner model. The effect of the Deborah number lambda, and the dimensionless retardation time epsilon on the hydrodynamic stability has been investigated and the range of values of these parameters for onset of overstability is obtained. The nonlinear hydrodynamic stability analysis has been done only for the range of these parameters where the exchange of stabilities occurs, that is the usual case with most polymeric liquids. It is confirmed that the rigid boundaries cause smaller convection amplitude and Nusselt number compared with the free boundaries and the convection amplitude and Nusselt number increase as the Deborah number lambda increases and as the dimensionless retardation time epsilon decreases. Moreover, the system tends to experience subcritical bifurcation as the value of lambda increases or the value of epsilon decreases, and the rigid boundaries have more tendency to cause subcritical bifurcation than the free boundaries when compared at the same values of elasticity parameters. The results of the present paper may be used to investigate the appropriateness of constitutive equations and their parameter values adopted for a given viscoelastic fluid.