If E/K is an elliptic curve over a number field and p is a prime number such that E has all its p-torsion defined over K, we show that the order of the p-torsion part of the Shafarevich-Tate group of E/L is unbounded as L ranges over degree p extensions of K. The proof uses the explicit period-index obstruction of C. O'Neil; indeed, we deduce the result from the fact that, under the same hypotheses, there exist infinitely many elements of the Weil-Chatelet group of E/K of period p and index p^2.