Aims:We calculate the bounds on the variation in the fine structure constant at the time of primordial nucleosynthesis and at the time of neutral hydrogen formation. We used these bounds and other bounds from the late universe to test the Bekenstein model. Methods: We modified the Kawano code, CAMB, and CosmoMC to include the possible variation in the fine structure constant. We used observational primordial abundances of D, ^4He, and ^7Li, recent data from the cosmic microwave background, and the 2dFGRS power spectrum, to obtain bounds on the variation in α. We calculated a piecewise solution to the scalar field equation of the Bekenstein model in two different regimes: i) matter and radiation, ii) matter and cosmological constant. We match both solutions with the appropriate boundary conditions. We performed a statistical analysis, using the bounds obtained from the early universe and other bounds from the late universe to constrain the free parameters of the model. Results: Results are consistent with no variation in α for the early universe. Limits on α are inconsistent with the scale length of the theory l being larger than the Planck scale. Conclusions: In order to fit all observational and experimental data, the assumption l > Lp implied in Bekenstein's model has to be relaxed.