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Accurate power characterization is important in computing platforms for several reasons ranging from poweraware adaptation to power provisioning. Power characterization is typically obtained through either direct measurements enabled by physical instrumentation or modeling based on hardware performance counters. We show, however, that linear-regression based modeling techniques commonly used in the literature work well only in restricted settings. These techniques frequently exhibit high prediction error in modern computing platforms due to inherent complexities such as multiple cores, hidden device states, and large dynamic power components. Using a comprehensive measurement framework and an extensive set of benchmarks, we consider several more advanced modeling techniques and observe limited improvement. Our quantitative demonstration of the limitations of a variety of modeling techniques highlights the challenges posed by rising hardware complexity and variability and, thus, motivates the need for increased direct measurement of power consumption.