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This paper considers the basic problem of "how accurate can we make Internet performance measurements". The answer is somewhat counter-intuitive in that there are bounds on the accuracy of such measurements, no matter how many probes we can use in a given time interval, and thus arises a type of Heisenberg inequality describing the bounds in our knowledge of the performance of a network. The results stem from the fact that we cannot make independent measurements of a system's performance: all such measures are correlated, and these correlations reduce the efficacy of measurements. The degree of correlation is also strongly dependent on system load. The result has important practical implications that reach beyond the design of Internet measurement experiments, into the design of network protocols.