Not too sure if this is a state or district thing, but I worked in a California school district where we kept records of every student, including their ethnicity. They were listed with a numerical code (700-white, 600-black, 500-hispanic, 400-asian, etc.) Every student that passed through our elementary, middle, or high school had a profile. I'm sure that if you were a bit suspect in your self-classification (and I'm not talking about the marginal "I can't tell what the heck he is" scenario) the bar could easily dig some of this up.
Also, whenever a minority student accepts an offer at a law school, one of the first people to hear about it are the student leaders of the certain group. Since every year the number of blacks or hispanics at a law school is between 0-15 (looking at NALP stats) each student receives a lot of attention. If you were clearly white as the fallen snow and your name went to the black law students group, I could only imagine the reaction when you show up to orientation. I heard of a student group in CA (at the undergraduate level) that called out someone in a scenario just like this and I'm pretty sure there was a stink at CLS about a female applicant who checked black but didn't look it. I'll try to find more info about it.
Finally, let us not underemphasize the importance schools place on your diversity statement and the effect your race has had in your life (in your PS for example) not to mention factors such as if you were a minority and the first one in your family to go to college, economic status, etc. It is here where you might find yourself in some trouble.