So I took a feat that allowed me to increase my base attack by one. This brought it from 10/5 to 11/6. Would that give me a third attack since it is actually raising my base attack?
Maybe you took a feat that raises your total attack bonus (probably only with a certain weapon/attack or in certain conditions), but surely not one that increases your Base Attack Bonus.
Iterative attacks are based on your BAB, so if there was some homebrew feat that you took that allowed you to increase your BAB to +11 then yes, you would have 3 iterative attacks on a full attack (+11/+6/+1).