The simple answer would be No, not in terms of killing power - generally the wider the bullet the better, and the heavier the bullet the better, without regard to velocity. Velocity is not always better - it usually is up to a point and then you get diminishing returns, or worse, reduced bullet performance. But a bigger, heavier bullet is almost always better, if not always better, in making quick kills, all other things being equal.
The more complicated answer is yes, all things are a tradeoff. The tradeoff is (1) more meat damage from the shock/hole of the bigger bullet, (2) more recoil, (3) more noise & muzzle blast, and (4) more ammo expense typically. That's why the most popular calibers represent such a good compromise - a sweet spot between the tradeoffs - calibers like .270, .30-06, etc. There's a good reason they are popular.