While the attempts currently in progress in several groups for the rigorous inclusion of dispersion interactions in density functional theory (DFT) calculations mature and evolve into practical methodology, we contribute to the debate on the applicability of current functionals to the calculation of weak interaction with a systematic investigation of a few, typical, weakly bound systems. We have used both pure DFT and a hybrid approach in which the total interaction energy is partitioned into two parts: (a) the dispersion energy which, in a first approximation is the contribution due to intermonomer correlations and (b) all other interactions. The first component is accurately obtained at all distances of interest by means of a well-known damped multipolar expansion of the dispersion energy while for the second component different approximations will be evaluated. The need to avoid double counting a fraction of the correlation energy when using the hybrid approach and the choice of the appropriate functional are also discussed. We consider four systems of increasing binding strength, namely the Ar-2 and Kr-2 dimers, the benzene dimer, the water dimer, and a few metal carbonyls. For pure DFT calculations we confirm the conclusion reached by others concerning (a) the strong dependence of the results on the choice of the GGA functional for dispersion-dominated interaction (noble gases and benzene) with the overall tendency to yield underbinding and (b) the relatively accurate, functional-independent, description for that DFT gives of water, which we attribute to the fact that this system is dominated by electrostatic interactions. For the carbonyls we find that DFT yields results which area again strongly dependent on the choice of the functional and show a tendency to give overbinding. Our hybrid method shows instead shortcomings only for the noble gases. The problem in this case is traceable to the well-known difficulties that all current functionals experience at medium-large intermonomer separations. The quality of the hybrid results improves markedly for benzene due to the large value of both dispersion and repulsive interactions at the equilibrium distance for this dimer, which makes the balance between the two, less delicate. Excellent results are also obtained for water (for the same reason as indicated above) and more significantly for the carbonyls where we find that dispersion contributes to the binding more than it could be guessed a priori. We do not claim to have found a general solution to this difficult problem, but we aim at providing a quantitative assessment to where the problems are pointing at directions from which a general solution may, eventually, emerge.