The curse of dimensionality has remained a challenge for a wide variety of algorithms in data mining, clustering, classification and privacy. Recently, it was shown that an increasing dimensionality makes the data resistant to effective privacy. The theoretical results seem to suggest that the dimensionality curse is a fundamental barrier to privacy preservation. However, in practice, we show that some of the common properties of real data can be leveraged in order to greatly ameliorate the negative effects of the curse of dimensionality. In real data sets, many dimensions contain high levels of inter-attribute correlations. Such correlations enable the use of a process known as vertical fragmentation in order to decompose the data into vertical subsets of smaller dimensionality. An information-theoretic criterion of mutual information is used in the vertical decomposition process. This allows the use of an anonymization process, which is based on combining results from multiple independent fragments. We present a general approach which can be applied to the k-anonymity, l-diversity, and t-closeness models. In the presence of inter-attribute correlations, such an approach continues to be much more robust in higher dimensionality, without losing accuracy. We present experimental results illustrating the effectiveness of the approach. This approach is resilient enough to prevent identity, attribute, and membership disclosure attack.