We generalise Theorem 1.4 of [2] and prove that for every complete extension {\bf T} of {\bf PA} and any $n\in\omega$ there exists a model for $\Sigma_n$--fragment of {\bf T} that is not extendable (that is, a model with no proper strong elementary end-extension.) This is accomplished using a model called $\Sigma_n$-atomic. This result can be interpreted as ``McDowell--Specker's Theorem does not hold for $\Sigma_n$-fragments of {\bf PA}''.