We study the connectivity of the excursion sets of additive Gaussian fields, i.e.\ stationary centred Gaussian fields whose covariance function decomposes into a sum of terms that depend separately on the coordinates. Our main result is that, under mild smoothness and correlation decay assumptions, the excursion sets $\{f \le \ell\}$ of additive planar Gaussian fields are bounded almost surely at the critical level $\ell_c = 0$. Since we do not assume positive correlations, this provides the first examples of continuous non-positively-correlated stationary planar Gaussian fields for which the boundedness of the nodal domains has been confirmed. By contrast, in dimension $d \ge 3$ the excursion sets have unbounded components at all levels.