Lots of variables. #1: Temperature  #2: Time  #3: Where it was shot.
   
    Above 40F the faster bacteria grows.
    The longer it lives the better, and  the better the shape it will be in.
    Keep in mind that the warmer the temperature the longer it will take for the insides of the deer to cool down. While the outside temp may be below 40 the inside of the deer is way above that, once it's dead then the gases start to form and bacteria starts to grow. The colder the outside temp the faster the interior of the deer would cool down.
    Under 30 degrees I would say you'll have no problem, 30 - 40 most likely no problem if it lived for awhile after the shot, above 40 then you will have to use common sense based on the bloating, gases, and where it was shot.
    A gut shot deer is always at higher risk of spoilage.

    When you field dress your deer, if the smell is something you wouldn't eat on meat from the store, then I wouldn't eat it from venison either.

Dave