I cannot think of many instances where a large company (more than 100 employees) was caught hiring illegals where the company wasn't defrauded by people using false ID's.
Here is an example of how their hands are tied:
http://www.nytimes.com/2009/05/05/us/05immig.html
The Supreme Court on Monday rejected a favorite tool of prosecutors in immigration cases, ruling unanimously that a federal identity-theft law may not be used against many illegal workers who used false Social Security numbers to get jobs.