For the most part depending on how good the quality(sample rate and bits) the original sound was recorded, will affect the final sound file. Yes a high bit sound will be better quality as long as it was recorded in that same sample rate and bit rate. Not to mention depending on what frequency range the microphone can pick up, also the speaker that is outputting the sound on the caller.
Lets say a typical microphone can record a range from 30Hz-16kHz, and the output amp and speaker can only go to 20khz, well a coyote can hear up to 45khz and bobcat up to 65khz. Now a bunch of that might not matter if the animal that was being recorded can't vocally go that high pitch (ie. bird/rabbit/hare/mouse/coyote).
Assuming the Sound was recorded in a high sample rate at 24 bit, I would want the recording to be on my caller in a 24bit. I think it helps but by no means would it a need or die on a caller.
As Technology gets better its easier to make what we consider higher quality sounds. Heck I have made good quality recordings of a $8 lapel mic that I purchased of amazon. Which sounds better then most of the 10-15yr old sounds that some of the major companies had/have.