For a smooth, stationary, planar Gaussian field, we consider the number of connected components of its excursion set (or level set) contained in a large square of area R2. The mean number of components is known to be of order R2 for generic fields and all levels. We show that for certain fields with positive spectral density near the origin (including the Bargmann-Fock field), and for certain levels ℓ, these random variables have fluctuations of order at least R, and hence variance of order at least R2. In particular this holds for excursion sets when ℓ is in some neighbourhood of zero, and it holds for excursion/level sets when ℓ is sufficiently large. We prove stronger fluctuation lower bounds of order Rα, α ∈ [1, 2], in the case that the spectral density has a singularity at the origin. Finally we show that the number of excursion/level sets for the Random Plane Wave at certain levels has fluctuations of order at least R3/2, and hence variance of order at least R3. We expect that these bounds are of the correct order, at least for generic levels.