I witnessed two inadvertent whistles this past week, one from a 1st year official and from a 2nd year official. Both were during a fumble when the players were piling up trying to recover a loose ball.
From my experience, most IW's occur during one of four plays:
1. During fumbles
2. When a player muffs a kick following a fair catch
3. After an official throws a flag and then brain-farts and blows his whistle
4. When an official blows his whistle thinking the FB has the ball during the triple option
And in my experience, play #1 (fumble) accounts for about 3/4's of IW's.
I think our new officials blow the whistle to "protect" the players (thinking the whistle will prevent players from piling on or being too aggressive in the pile trying to recover the loose ball). The whistle should never be blown to "protect" players following a fumble or fair catch signal. We put dirty air in our whistles when forward progress is stopped and the runner is being pushed back, but that whistle indicates the play has ended by rule.
During our pre-game, I try to remember to tell our crew not to blow the whistle when there is a pile following a fumble, but to get to the pile quickly. The first to get there is the "digger." Others are "peelers," selecting individual players without the ball and using phrases like "you don't have it" or "it's not your ball." Then we have a few officials that stay away from the pile after stopping the clock and use their "big eyes" to watch for buffoonery.
We tell our new officials it's okay to not have a whistle on a play if no one can see the ball down in player possession. We tell our new officials to avoid blowing the whistle if the ball is downed outside their primary area (unless they are the only one who can see the ball down). We tell our new officials not to "mirror" another official's whistle. But it still takes a few years to master having a slow/patient whistle.