Is "disconfirm" a word?

Yes. Some dictionaries include it, like m-w.com:

to deny or refute the validity of

And TFD:

1. (tr) (of a fact or argument) to suggest that a hypothesis is wrong or ill-formulated

However, synonyms like refute and disprove, and near-synonym rebut, are much more popular, as Google Ngrams Viewer shows:

Yes, but it is used only rarely, according to a Google search.

• When the communicator confirmed the subjects' expectancies, he was judged more biased than when he disconfirmed them. - ADV EXPERIMENTAL SOCIAL PSYCHOLOGY
• This suggests that Nash treated his delusion as a hypothesis that he disconfirmed. - The Measure of Madness: Philosophy of Mind, Cognitive Neuroscience, and Delusional Thought
• We have also seen the same pattern at work in A through O, where the only direct interlocutor, so to speak, was Socrates himself, as he disconfirmed one meaning of wise... - Does Socrates Have a Method?: Rethinking the Elenchus in Plato's Dialogues

It seems to be used more in analytical writing.

You're in a unique position to confirm or deny that suspicion...

seems more natural to me. It's hard to "refute" a suspicion...

Considering that its first Google Books search match of the twentieth century did not occur until 1913, and its first nonlegal matches not until 1938, disconfirm has etched a fairly impressive-looking Ngram chart between 1900 and 2005. For purposes of comparison (and scale), I matched disconfirm (blue line) against the familiar verb discombobulate (red line):

The 1913 instance of disconfirm is from Laws, Joint Resolutions, and Memorials Passed by the Legislature of the State of Nebraska at the Thirty-third Session (April 26, 1913):

{Same [amendment], determination; bonds, legality, investment.} Upon the hearing of such special proceeding the court shall have power and jurisdiction to examine and determine the legality and validity of, and approve and confirm or disapprove and disconfirm each and all of the proceedings for the organization of said district under the provision of this act, from and including the petition for the organization of the district, and all other proceedings which may affect the legality or validity of said bonds and the order of the sale and the sale thereof.

In 1915, dazzled by the syntactical perfection of Nebraska's statutory language, Oklahoma adopted almost identical language (including the word disconfirm) for its statutory code on the same subject, in State of Oklahoma Session Laws of 1915. Regrettably no other state has yet joined Nebraska and Oklahoma on this particular bandwagon, but there's still plenty of room on board.

The next fields to succumb to the charms of disconfirm seem to have been education and psychiatry (both in 1938) and then symbolic logic (in 1940). In subsequent years, disconfirm gained a following in various other academic fields. If you have any lingering doubts as to its legitimacy, perhaps this next example will serve to discorroborate them. From J.T. Roberts, "Foreign Language Learning in Anglosaxony: An Attempt at a Culturo-Psychological Perspective" (1978), published in The Modern Language Journal (December 1979) [combined snippets]:

In November 1976 there took place at the behest of the professional associations for language teachers a conference with the very watchword in its title: "The Crisis in Modern Languages." It is as yet too early to confirm or disconfirm the worst fears expressed, but the concern is ongoing and has recently preoccupied a ... [remainder of sentence unreadable].

To similar effect is Anna Wierzbicka, English Speech Act Verbs: A Semantic Dictionary (1987) [combined snippets]:

Discussion: Denying involves, above all, saying of something 'this is not true'. Somebody else says that X (either the addressee or a third party), and the speaker responds by saying 'this is not true'. In fact, it might seem that saying 'this is not true' is all there is to it. One might expect that this is all that a logician or a logic-oriented linguist would want to say about deny. In natural language, however, things are rarely if ever as simple as that. Saying 'this is not true' is also involved in disconfirming, and yet deny and disconfirm differ considerably from one another.

To begin with, disconfirm sounds factual and dispassionate, whereas deny tends to sound passionate and personally involved. One wouldn't normally deny that Oslo is the capital of Sweden or that Marco Polo lived in the fourteenth century. Furthermore, disconfirm seems to be restricted to observable facts, accessible to anybody who happens to be a witness; denying, on the other hand, can also apply to inner facts, which could only be known to the experiencer himself. Thus, if someone says that I am jealous of somebody I could deny it (because who should know if not me), but it would be odd for me, and even odder for somebody else, to confirm it, or disconfirm it.

Clearly, disconfirm plays a role in academic/scientific discourse that isn't going to disappear anytime soon. It appears that Brandt in The Big Lebowski, like Roberts in The Modern Language Journal, was making an attempt at a culturo-psychological perspective. As such, disconfirmation was certainly within his aegis, and, indeed, very likely the least of his problems.