The large wealth and consumption inequality in the U.S. is usually attributed to two market frictions: debt constraints and incomplete markets. Recent literature has argued that debt constraints are the critical friction while market incompleteness plays only a secondary role. We evaluate the independent role of debt constraints versus market incompleteness to explain U.S. inequality. We introduce full insurance opportunities in a standard model of inequality along the lines of Aiyagari (1994). Debt constraints are the only friction in such model. We find that for a quite standard calibration of the income process, that of Heaton and Lucas (1996), debt constraints alone can explain none of the observed inequality. The reason is that the U.S. capital stock would be enough to secure all required contingent debts if markets were completed. Using various non-standard calibrations, we find that debt constraints can play an important role to explain inequality but still market incompleteness remains as the main friction. In particular, debt-constrained models cannot account for the large wealth dispersion and wealth concentration in the top tail of the distribution in the U.S.