Are we going to split hairs here? :-)
Seriously, I think the date on the check is just one more example of the no-frame-of-reference open-endedness that permeates the film.
The first conflict with the Eye-rackies began in the summer of 1990 as was, therefore, Bush's "aggression" comment (which was made on August 5, 1990). The Dude writes the check for Sept. 11, 1991, more than a year later. Marty says it's already the 9th...is he referring to August 9? Sept. 9? While either is plausible (maybe the Bush comment is tape being played on a program in September), it doesn't matter in some regard because the main point is the Dude is a year off, intentionally or not.
For me, this ambiguity is intentional on the Coen Brothers' part because it fits right in there with the rest of the movie's ambiguity around the meaning of words, identities, etc... but that's just, like, my opinion, man.
BTW, in the interest of full disclosure, I'm Dwayne, one of the co-authors of the book. Not trying to scam anyone here. :-)