Can a binomial distribution have negative x values? The Poisson distribution makes a big deal that its great for modeling "Counts" of stuff (aka counts of monthly users) because its a non negative discrete distribution.
Well, aren't Binomial and Bernoulli distributions also non negative?
I can't think of any discrete distributions that can have negative x values. (Normal distribution i believe is a continuous distribution so it doesn't count)

The binomial distribution is a distribution for a sum of Bernoulli trials: the sum of zeros and ones is non-negative. But it is not true that all discrete distributions are non-negative.
For a trivial example, say that $X$ follows a Poisson distribution and you create another variable $Y = -X$. $Y$ would be discrete and all its values would be smaller than or equal to zero by definition. The distribution for $Y$ does not have a name (it's like a Poisson distribution, just look at the $-y$ values), but as noted in the comments, there are named examples like the Skellam distribution.
