No. Nagata gave an example of an linear action of $\mathbb{G}_a^k$ on a complex vector space $V$ such the ring of invariants $\mathrm{Sym}^{\bullet}(V)^{\mathbb{G}_a^k}$ is not finitely generated. On the other hand, Wietzenbock proved that, if $\mathbb{G}_a$ acts linearly on a complex vector space, then $\mathrm{Sym}^{\bullet}(V)^{\mathbb{G}_a}$ is finitely generated. The invariants for $\mathbb{G}_a^k$ can be written as the intersection of the invariants for $k$ different actions of $\mathbb{G}_a$.
References: Nagata, Wietzenbock, lecture notes by Nagata which prove both theorems.