Technically speaking, Nihilism is only impossible if you use one specific definition for the term. Unfortunately, a conditionally impossible "ism" just doesn't have the same punch in a title. That said, Nihilism has all too often been used as a smear. It is usually aimed at atheists but has been leveled at various other individuals and groups.
Before going any further it is essential to establish a basic definition, or set of definitions for the term in question. According to the Penguin English Dictionary:
a view that rejects all values and beliefs as meaningless or unfounded.
the philosophical view that nothing has real existence; extreme scepticism.
(often Nihilism) the doctrine of a 19th-cent. Russian revolutionary party that social conditions are so bad as to make destruction desirable for its own sake.
terrorism or anarchy
nihilist noun and adj nihilistic /-ʹlistik/ adj [German Nihilismus, from Latin nihil nothing]."
The majority of theologians, apologists, and rabid right-wingers who choose to wield the term like a cudgel usually are using 1, 2, or both. Their usage of the term tends to be related to the absurd notion that if an individual does not believe in God then they cannot possibly have any beliefs at all. It is in the sense of a complete lack of beliefs that Nihilism becomes impossible.
Everyone has beliefs. In practical terms you cannot function on a day to day level without beliefs. Believing in nothing is an innate contradiction.* How could you do anything? Why would you even bother trying to do anything? Even if an individual thinks they have no beliefs they have just contradicted themselves, They have in effect expressed a believe in nothing. If you choose to interpret the meaning to pertain to profound beliefs or a set of beliefs that can be used as an actual "ism", a sort of philosophical guide, then that would not be impossible. However, that is an interpretation and is not reflective of how the term is commonly used.
*Believing in the concept of "nothing" is not what I am talking about. Nothing as a mental construct is quite useful. Complete emptiness (i.e. oblivion, void, absolute zero....) has spurred numerous debates, discussions, and explorations.