QNAME minimization is an extension to the DNS protocol, designed to allow DNS resolvers to prevent disclosure of DNS activity beyond that which is necessary for resolution. Since it was originally proposed in 2014, QNAME minimization has been incorporated into most of the well-known DNS resolvers. But the question remains: how effective is QNAME minimization at preserving privacy in practice? We answer that question by creating a model that defines DNS privacy roles and quantifies information leakage to third parties. We apply that model to DNS query data from a large university. We observe that QNAME minimization adds modest privacy gains and suggest that its benefits be considered alongside its costs.