I know a couple of years ago they stopped serving at 11:00 AM. I remember this specifically since we got to the resort at 10:50 and just made breakfast. However, I know the Grand Floridian Cafe now serves until 11:30 with no break in service between breakfast and lunch. That may now also be the case at Kona Cafe.
If you go to the Disney dining reservations page and enter Kona Cafe, it'll tell you the time it is serving breakfast. I don't know if it varies according to the day, but right now breakfast is 7:30 - 11:15.