We investigate the conditional full support (CFS) property, introduced in Guasoni et al. (2008a), for Gaussian processes with stationary increments. We give integrability conditions on the spectral measure of such a process which ensure that the process has CFS or not. In particular, the general results imply that, for a process with spectral density f such that f(λ) ∼ c 1λp e−c2λ q for λ → ∞ (with necessarily p < 1 if q = 0), the CFS property holds if and only if q < 1.