Non-estimable linear parametric functions

Consider the linear model $Y_{n\times1}=X_{n\times p}\theta_{p\times1} +\epsilon_{n\times1}$ with $\mathbb{E}(\epsilon)=0$ and $\operatorname{Cov}(\epsilon)=\sigma^{2}I_n$. If $C$ is a $p\times1$ vector such that $C'\theta$ is not estimable, then which statement is always true?