This paper derives upper bounds on heat transport in an eight-mode model of Rayleigh-Bénard convection. It uses both the background method and an optimal-control formulation in which the velocity field acts as a control variable. The two approaches lead to the same transport estimate and extend ideas previously applied to the three-variable Lorenz system. Unlike the Lorenz case, the best bound obtained does not always appear to correspond to an exact trajectory of the original equations. Readers get a comparison of two bounding techniques and an explanation of why a mathematically admissible transport limit need not be dynamically attainable.