One sharp example of this phenomenon is the so-called Reeh-Schlieder (RS) theorem. In this presentation, I will show explicitly how the RS theorem uses the assumption of relativistic causality to derive a highly nonlocal conclusion. In particular, I will focus my discussion around explicating two claims that have been made about the RS theorem: (1) Simon Saunders claims that the RS theorem is a "purely relativistic result." I will show that while in one mathematically precise sense, Saunders' claim is true -- viz., the RS theorem fails when we pass to the non-relativistic limit (c -> infinity) -- in the strictest sense it is false, since a modified version of the RS theorem continues to hold in nonrelativistic, Galilei-invariant quantum field theory. (2) According to Irving Segal, the RS theorem (if its premises were true) would entail that, "the entire state vector space of the field could be obtained from measurements in an arbitrarily small region of space-time!" I will show the precise sense in which Segal's claim is true, and I will argue that the resulting nonlocality is not -- as Segal supposes -- "at variance with the spirit of relativistic causality."