The author presents several results which generalize some known theorems on the existence of nonmeasurable (in various senses) sets of the form \(X+Y\). The main results: If \(\mathcal I\) is a \(\sigma\)-ideal of subsets of \(\mathbb R\) and \(\mathcal B\) is a family of sets containing \(\mathcal I\) such that every set \(B\in {\mathcal B}\setminus {\mathcal I}\) contains a non-empty perfect set, then: (1) For every set \(A\subset {\mathbb R}\) such that \(A+A\not\in {\mathcal I}\) there exists a set \(X\subset A\) such that \(A+A\not\in {\mathcal B}\); (2) For every sets \(A,B\subset {\mathbb R}\) with \(A+B\not\in {\mathcal I}\) there exist \(X\subset A\), \(Y\subset B\) such that \(X+Y\not\in {\mathcal B}\). In the case when \(\mathcal I\) is the ideal of Lebesgue measure null sets (meager sets) and \(\mathcal B\) is the algebra of Lebesgue measurable sets (sets with the Baire property) then: (3) If \(A,B\in {\mathcal B}\setminus {\mathcal I}\) then there exist \(X,Y\in {\mathcal I}\) such that \(X\subset A\), \(Y\subset B\), and \(X+Y\not\in {\mathcal B}\); (4) If \(A\in {\mathcal B}\) and \(A+A\not\in {\mathcal I}\) then there exists \(X\in {\mathcal I}\), \(X\subset A\) such that \(X+X\not\in {\mathcal B}\). Finally, the author shows that for each uncountable set \(A\subset{\mathbb R}\) there is \(X\subset A\) such that \(X+X\) is not analytic. On the other hand, there exists a perfect set \(P\) such that for any pair of Borel sets \(A, B\subset P\) the set \(A+B\) is Borel.