Household surveys often suffer from nonresponse on variables such as income, savings or wealth.Recent work by Manski shows how bounds on conditional quantiles of the variable of interest can be derived, allowing for any type of nonrandom item nonresponse.The width between these bounds can be reduced using follow up questions in the form of unfolding brackets for initial item nonrespondents.Recent evidence, however, suggests that such a design is vulnerable to anchoring effects.In this paper Manski's bounds are extended to incorporate the information provided by the bracket respondents allowing for different forms of anchoring.The new bounds are applied to earnings in the 1996 wave of the Health and Retirement Survey.The results show that the categorical questions can be useful to increase precision of the bounds, even if anchoring is allowed for.