...And sometimes you're the bug.
Straight from AR 670-1:
Authorization to wear a shoulder sleeve insignia indicating former wartime service applies only to soldiers who are assigned to U.S. Army units that meet all the following criteria. Soldiers who were prior members of other Services that participated in operations that would otherwise meet the criteria below are not authorized to wear the SSI–FWTS. Wear is reserved for individuals who were members of U.S. Army units during the operations.
To 99% of the population, this means nothing, but the naked right sleeve on the ACU is an instant credibility hit for the wearer. I came across many different answers to this question via Google searches and looking at Internet forums, but this seems pretty durned straightforward. No one cares whether you rate a Combat Action Ribbon, how many Campaign Medals you have, whether you heard real IEDs with your own two ears, or saw mortars come in with your own two eyes, or heard AK-47s fired in anger -- if you weren't in the Army, you don't rate the patch.
Just remind me not to get too cagey when someone pats me on the head and says, "Don't worry, you'll understand once you see the elephant."