The potential curves of low-lying singlet states of Be2 are computed in the hierarchy of coupled-cluster (CC) models: CCS, CC2, CCSD, and CC3 from the ground-state energy and linear response excitation energies. The results are compared with full configuration interaction (FCI) values. For single-excitation dominated transitions the convergence of the excitation energies toward FCI is smooth with a reduction in the error of about a factor of 3 at each level of the CC hierarchy. This reduction is only seen from CCSD to CC3 for the double-excitation dominated transitions. CC3 gives accurate potential curves and spectroscopic constants of the excited states dominated by single excitations. For the double-excitation dominated states the CC3 equilibrium distances and vibrational frequencies agree reasonably well with FCI, while the bonding energies and the 0-0 excitation energies have larger errors. We analyze the influence of the frozen-core approximation, basis-set effects and the counterpoise correction on the ground-state energy, the excitation energies, and molecular constants. Considering all these effects, we provide an improved estimate of the calculated properties.