We give a characterization for those stable theories whose $\omega_{1}$-saturated models have a "Shelah-style" structure theorem. We use this characterization to prove that if a theory is countable, stable, and 1-based without dop or didip, then its $\omega_{1}$-saturated models have a structure theorem. Prior to us, this is proved in a paper of Hart, Pillay, and Starchenko (in which they also count the number of models, which we do not do here). Some other remarks are also included.