A version of the Hilbert-Mumford criterion states the following: Let $G$ be a linearly reductive group and $V$ a representation of $G$ over a field $k$ (alg. closed, char. zero). Suppose that $y \in \overline{Gx} - Gx$. Then, there is a one-parameter subgroup $\lambda : k^\times \to G$ such that $$ \lim_{t\to 0} \lambda(t)x \in \overline{Gy}. $$

My question is: Is there an example where every one parameter subgroup misses the orbit of $y$? I.e., is there an example where, for every $\lambda: k^\times \to G$ $$ \lim_{t\to 0} \lambda(t)x \in \overline{Gy} \implies \lim_{t\to 0} \lambda(t)x \in \overline{Gy}-Gy? $$ If $G$ is a torus the answer is "no". What if $V$ is replaced by a more general scheme $X$ that is not itself a representation?