Advancements in targeting technology have allowed firms to engage in more precise targeting based on several aspects of consumers' preferences. Exposed to more targeted ads, consumers are becoming increasingly aware of being targeted and respond accordingly. This paper provides a theoretical analysis of multidimensional targeting under which consumers can draw inferences about multiple components of their utility from the advertised product. We show that the firm can be worse off under multidimensional targeting than under single-dimensional targeting, in which the firm targets consumers based only on a single component of their utility. This is because, with multidimensional targeting, targeted consumers may face greater uncertainty about which specific dimension(s) they can expect to enjoy the advertised product. Therefore, they may be less willing to exert a costly effort of clicking the ad and purchasing the product. When this result holds, the firm may want to adopt a single-dimensional targeting strategy. However, we show that the firm cannot credibly commit to such a strategy once given access to multiple dimensions of customer data. Interestingly, a higher unit cost of advertising can mitigate the firm's commitment problem for utilizing customer data and, thus, increase the firm's profit. Moreover, the firm can sometimes lower the price to recover some of, but not entirely offset, the drawbacks of multidimensional targeting. We discuss the implications of our results regarding the current practice of targeted advertising and data privacy protection policies.