Dense lineability and spaceability in certain subsets of $\ell_{\infty}$

We investigate dense lineability and spaceability of subsets of $\ell_\infty$ with a prescribed number of accumulation points. We prove that the set of all bounded sequences with exactly countably many accumulation points is densely lineable in $\ell_\infty$, thus complementing a recent result of Papathanasiou who proved the same for the sequences with continuum many accumulation points. We also prove that these sets are spaceable. We then consider the same problems for the set of bounded non-convergent sequences with a finite number of accumulation points. We prove that such set is densely lineable in $\ell_\infty$ and that it is nevertheless not spaceable. The said problems are also studied in the setting of ideal convergence and in the space $\mathbb{R}^\omega$.