Fibre of GIT morphism Let $ V $ be an affine variety (over $ \mathbb C$) with an action of a reductive group $ G$.  I would like to consider the morphism $$ \pi : V \rightarrow V // G = Spec \, \mathbb C[V]^G $$
Let $ v \in V $.  Assume that the orbit $ Gv $ is closed in $ V $. Assume also that the stabilizer of $ v $ in $ G $ is finite.  
Question:
Is the following true?  What additional hypothesis should I place on $ v $ in order to ensure that the following is true?

The scheme-theoretic fibre $ \pi^{-1}(\pi(v)) $ equals $ G v $.

I looked in Mumford's book, but I could not find this.
Example:
Here is an example where this does work.  Suppose that $ G = SL_k $ and $ V = Hom(\mathbb C^k, \mathbb C^n) $ and let $ v $ be an injective map.  Then $ V // G $ is the variety of pure $k$-tensors (the cone on the Grassmannian).
 A: I am just recording what was said in the comments so this question does not appear completely unanswered.
Let $\pi_X:X\to X//G$ be the GIT quotient of an affine variety over $\mathbb{C}$ by a reductive group $G$.  WLOG assume the action is effective.
First, a point is properly stable if its orbit is closed and it has finite stabilizer.  The locus of properly stable points is Zariski open.  Then since each fibre $\pi_X^{-1}(\pi_X(x))$ is a union of orbits, this union contains a unique closed orbit, two such orbits are in the same fibre if and only if their closures intersect, and such intersections can be detected by 1-parameter subgroups, we can conclude that if $x$ is properly stable then $\pi_X^{-1}(\pi_X(x))$ is set-theoretically the orbit $Gx$.
Please see Stability of Affine G-varieties and Irreducibility in Reductive Groups by Casimiro and Florentino as a reference.
As pointed out by Jason Starr in the comments, the fibre is not generally scheme-theoretically the orbit however. A counter-example is the action of $\mathbb{G}_m$ on $\mathbb{A}^2\times \mathbb{G}_m$ by $s\cdot((x,y),t)=(sx,sy,s^{-n}t)$ for $n>1.$  As noted by the OP, this is apparent even for $n=2$.
We now refer to the Luna Slice Theorem; see Luna’s slice theorem and applications by Drézet.  Let $V$ be a slice at a properly stable point $x$, and let $\pi_V:V\to V//S$ be the corresponding quotient where $S$ is the stabilizer of $x$ (necessarily a reductive subgroup).  Then there is an isomorphism: $$G\times_S \pi^{-1}_V(\pi_V(x))\cong \pi_X^{-1}(\pi_X(x)).$$ 
So, the fibre is the scheme-theoretic orbit if it is smooth which, by the Chevalley-Shephard-Todd Theorem, occurs if and only if the stabilizer is generated by pseudoreflections.
