Yes. I've cut venison the same day it was shot with no problems. As you know body mass is made up primarly of water, much of which is in the blood. The more water (blood / moisture) that is taken out of the meat the dryer and tougher it can become, therefore I have never recommended hanging deer for extended periods.
    If you want to illustrate this concept, buy two t-bones or ribeyes, cook one to 145 degrees (medium rare) and one to 165 degrees (well done). Then taste the difference. The majority of the population (not all)  will go for the 145 degree steak as it still retains the mositure and is more tender.

Dave