Let X be an uncountable Polish space and let $\mathcal {I}$ be an ideal on $\omega $. A point $\eta \in X$ is an $\mathcal {I}$-limit point of a sequence $(x_n)$ taking values in X if there exists a subsequence $(x_{k_n})$ convergent to $\eta $ such that the set of indexes $\{k_n: n \in \omega \}\notin \mathcal {I}$. Denote by $\mathscr {L}(\mathcal {I})$ the family of subsets $S\subseteq X$ such that S is the set of $\mathcal {I}$-limit points of some sequence taking values in X or S is empty. In this article, we study the relationships between the topological complexity of ideals $\mathcal {I}$, their combinatorial properties, and the families of sets $\mathscr {L}(\mathcal {I})$ which can be attained. On the positive side, we provide several purely combinatorial (not depending on the space X) characterizations of ideals $\mathcal {I}$ for the inclusions and the equalities between $\mathscr {L}(\mathcal {I})$ and the Borel classes $\Pi ^0_1$, $\Sigma ^0_2$, and $\Pi ^0_3$. As a consequence, we prove that if $\mathcal {I}$ is a $\Pi ^0_4$ ideal then exactly one of the following cases holds: $\mathscr {L}(\mathcal {I})=\Pi ^0_1$ or $\mathscr {L}(\mathcal {I})=\Sigma ^0_2$ or $\mathscr {L}(\mathcal {I})=\Sigma ^1_1$ (however we do not have an example of a $\Pi ^0_4$ ideal with $\mathscr {L}(\mathcal {I})=\Sigma ^1_1$). In addition, we provide an explicit example of a coanalytic ideal $\mathcal {I}$ for which $\mathscr {L}(\mathcal {I})=\Sigma ^1_1$. On the negative side, since $\mathscr {L}(\mathcal {I})$ contains all singletons, it is immediate that there are no ideals $\mathcal {I}$ such that $\mathscr {L}(\mathcal {I})=\Sigma ^0_1$. On the same direction, we show that there are no ideals $\mathcal {I}$ such that $\mathscr {L}(\mathcal {I})=\Pi ^0_2$ or $\mathscr {L}(\mathcal {I})=\Sigma ^0_3$. In fact, for instance, if $\mathcal {I}$ is a Borel ideal and $\mathscr {L}(\mathcal {I})$ contains a non $\Sigma ^0_2$ set, then it contains all $\Pi ^0_3$ sets. We conclude with several open questions.