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We address the important question of the extent to which random variables andvectors with truncated power tails retain the characteristic features of randomvariables and vectors with power tails. We define two truncation regimes, softtruncation regime and hard truncation regime, and show that, in the softtruncation regime, truncated power tails behave, in important respects, as ifno truncation took place. On the other hand, in the hard truncation regime muchof "heavy tailedness" is lost. We show how to estimate consistently the tailexponent when the tails are truncated, and suggest statistical tests to decideon whether the truncation is soft or hard. Finally, we apply our methods to tworecent data sets arising from computer networks.

Understanding heavy tails in a bounded world or, is a truncated heavy tail heavy or not?