Generally, when folks refer to "overkill" or "too much gun", they're referring to two factors: More power than is needed for the type of game, and the amount of recoil which commonly accompanies such cartridges.
For deer and cougar, then, even an '06 is at the upper end of the spectrum. For practicality, I'd probably not use less than a .243, but that's somewhat argumentative. At any rate, something in the power-neighborhood of a 7mm08 wiould work quite well.