Bose-Einstein correlations between like-charge pions are studied in hadronic final states produced by e+e- annihilations at center-of-mass energies of 172 and 183 GeV. Three event samples are studied, each dominated by one of the processes W+W−→qq¯¯¯ℓν¯¯¯ℓW+W−→qq¯ℓν¯ℓ , W+W−→qq¯¯¯qq¯¯¯W+W−→qq¯qq¯ , or (Z0/γ)∗→qq¯¯¯(Z0/γ)∗→qq¯ . After demonstrating the existence of Bose-Einstein correlations in W decays, an attempt is made to determine Bose-Einstein correlations for pions originating from the same W boson and from different W bosons, as well as for pions from (Z0/γ)∗→qq¯¯¯(Z0/γ)∗→qq¯ events. The following results are obtained for the individual chaoticity parameters λλ , assuming a common source radius R: In each case, the first error is statistical and the second is systematic. At the current level of statistical precision it is not established whether Bose-Einstein correlations, between pions from different W bosons exist or not.