Realistically that is about it 2 to 2.5 inches.
Think about it...you need some space above where the brim meets the cap lets say about 1/2 inch on the high side.
Then a 2 inch high logo
then some clearance for the presser foot within the cap frame, say 1/8 (real tight) at bottom
and lastly the cap frame itself
SOOOOOOO 1/2 in from the brim to the inside of the cap hoop/frame 1/8 in clearance for the presser foot, 2 1/4 in high design, you are just about at 3 inches
Couple this with the curvature of the cap style and you are pretty much there.
Here is a pretty good article the sews it up pretty well, pay special attention to the paragraph
HOOPING CORRECTLYhttp://impressions.issshows.com/commercial-embroidery/Cap-Embroidery-Made--3998.shtmlmooseman