Can eating a ground beef patty lead to illness?

Yes, particularly if it isn't thoroughly cooked. The major concern is the pathogen E.coli, which can cause hemorrhagic colitis in humans.  Although meat processors must have food-safety protocols in place to ensure all packaged products are pathogen-free, disease outbreaks involving E. coli and ground beef can--and do--occur.
 
Most pathogens, E.coli included, generally reside only on the surface of meat.  However, because ground beef undergoes grinding, if pathogens are present, they can be spread throughout the product.  Cooking ground beef to an internal temperature of 160°F is a must, but it may not be sufficient to disable all pathogens.

A 2009  New York Times article quotes food safety expert Dr. Jeffrey Bender as saying, “Ground beef is not a completely safe product.”  Why not?  First, ground beef is commonly made from different parts of the animal and, at least in some cases, with product from different slaughterhouses.  Both of these circumstances increase the odds of contamination.  Furthermore, grinders may use scraps and fatty trimmings that are more likely to have come in contact with feces  (which carry E.coli).  The  FDA has encouraged  but not required grinders to test ingredients before combining them, when it would be easier to detect pathogens.  Suppliers do check ground beef for bacteria but may not catch all of them. 
The result: every year, tens of thousands of people are sickened by the virulent strain of E.coli known as 0157:H7.  In the summer of 2009, contamination led to the recall of beef from nearly 3,000 grocers in 41 states.  Nevertheless, considering the great quantity of ground beef consumed by Americans, your chances of getting sick from it may be rather small.  Still, no one is happy with the current level of illness caused by this product.     

In addition, ground beef  may be subject to bacterial growth if not stored at a temperature of 40°F or less or if stored at an acceptable temperature for a prolonged period of time. 
 
Source(s):
Meatingplace Magazine "Hide Away" (Note: Readers must be members of the following source in order to access information.)
The New York Times. "E. Coli Path Shows Flaws in Ground Beef Inspection" 
http://www.nytimes.com/2009/10/04/health/04meat.htm/o4meat.html?_r=18emc=etal