Non-random item nonresponse makes identification of parameters problematic. Such nonresponse can occur with respect to both dependent and conditioning variables. A method often used to reduce nonresponse is that of adding unfolding brackets as follow up to open-ended questions. With these, initial non-respondents can provide additional (incomplete) information on the missing value. However, recent studies suggest that responses to unfolding brackets can lead to a type of bias as a result of 'the anchoring effect'. In this paper, bounding intervals of the type as presented in Horowitz and Manski (1998) are extended to incorporate information provided by bracket respondents while allowing for different types of anchoring, and, therefore,accounting for significant nonresponse in the conditioning set. The theoretical framework is illustrated with empirical evidence based on the 1996 wave of the Health and Retirement Study.