Zero-*inflated* negative binomial models don't assume that *all* the zeroes come from the Bernoulli process; some may come from the negative binomial process. Toss a coin & write down zero if it's tails. If it's heads then start tossing again & write down the number of tails until you have three, say, heads. There are two different reasons for writing down zero so you can't separate the data into two parts for different models.