It's probably the least important factor for them, as I'd put rainfall, food and predation all above hunting pressure. Rainfall is the only one of those we can't control. We can have burns to promote quality native grasses, kill every predator that moves (except those darn hawks), and limit the amount of hunting pressure.
I just figure that if you have 4 hunters per weekend shooting limits of birds each day over the course of 18 weekends in a season, its easier on your population to take out 360 (5 bird limit) birds than 1080 (15 bird limit).
**Disclaimer: Although we would love to hunt every weekend, that doesn't happen.