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)
 A: 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.
A: A less intuitive but important example for a discrete distribution with negative support would be the distribution of (unit or dollar) sales for a particular product at a given point of sale, conditional on various predictors like the day of the week, time of the year, promotions etc.
This is discrete because most products are sold by units (and therefore, dollar sales are also discrete). It has potential negative support because of product returns. Anecdotally, US consumer electronics retailers have quite a problem with big screen TVs that are bought right before the Super Bowl and returned for a full refund right after the game.
A: Poisson, binomial, Bernoulli, negative binomial, etc. are just model distributions - that is distributions that are analytically tractable and/or can be derived under rather simple assumptions. One could thus reformulate the question as:

Are there known model discrete distributions with a support containing negative numbers?

Then the question becomes obviously about how we define these discrete distributions - the most well-known ones are defined with support on non-negative integers. Specifically the binomial has support
$$
k\in\{0,1,...,n\}$$
As @Tim pointed out in their answer, we can easily define a distribution with negative support, using the one with positive support.
This brings us to the phenomena that we actually describe: thinking of a problem where negative counts arise naturally could suggest a distribution (which is not necessarily a known and/or model one.) However, many real counts are by nature positive - like the mentioned counts of monthly users (unless we explicitly shift the origin, just as in @Tim answer.)
Remarks:

*

*Wikipedia has a list of discrete probability distributions, many of which are the distributions on integers. On a brief glance none of them has support extending to negative numbers.

*A similar question: Can the discrete variable be a negative number?
