You don't hear Republicans criticizing this system -- and it was a Republican president, Richard Nixon, who introduced it in the federal government in 1970. It quickly spread to academia and corporate America.
People who classify themselves as approved minorities get into schools and get jobs that they wouldn't if they classified themselves as white. Not surprisingly, some people, perhaps including Warren, game this system.
The original justification was that this would overcome the disadvantages that American blacks endured during decades of slavery and segregation. That made sense to many people at the time. Those disadvantages were real, and most Americans wanted to be fair.
But the extension of minority status to other groups and the perpetuation of racial preferences for nearly half a century since the abolition of legal segregation mean that there is increasingly little correlation between membership in the favored categories and genuine disadvantage.
Black leaders lament that black college admissions increasingly favor affluent blacks (Barack Obama has made this point) and recent immigrants from Africa. Hawaii Sen. Daniel Akaka sought to separate Pacific Islanders from Asians, since the former are more disadvantaged.
One solution would be to ban self-description and come up with rigorous definitions of race, like Louisiana's post-Reconstruction racial code or South Africa's apartheid laws. But those don't seem like very attractive models.
An alternative is to ditch racial quotas and preferences altogether. Retiring Democratic Sen. Jim Webb has made a proposal for something like this.
The strongest argument for this is not that some whites (and Asians) get passed over; these individuals will probably do fine nonetheless. The strongest argument against the system is that it casts a pall of illegitimacy over the genuine achievements of the intended beneficiaries.
In the meantime, what may undermine racial quotas and preferences most effectively is ridicule. For isn't the idea that the blond, blue-eyed Elizabeth Warren suffered some terrible disadvantage and is in need of special preference because she is one-thirty-second Cherokee just laugh-out-loud funny?