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][1], [Wietzenbock][2], [lecture notes by Nagata][3] which prove both theorems.


  [1]: http://www.ams.org/mathscinet-getitem?mr=116056
  [2]: http://link.springer.com/article/10.1007%2FBF02547779
  [3]: http://www.math.tifr.res.in/~publ/ln/tifr31.pdf