What to do if you suspect combinations of two distributions? My techniques so far is old school (got my degree in 1980s) - regression, crosstab, ... at best Poisson and negative binomial.  SPSS/SAS and now R.  Usually just run and report ... 
I encounter now real data where the count results look as a sharp peak in 0 and declining then another distribution with quite a normal around some positive figures.  I said that we have 2 populations joint together.  
What to do?  No firm idea.  Should I just ignore the graph (would report it) and just run based on the hypothesis hand down and report back it would have some up to R2 30%.  t test sig. for some factors. But is that it?  I am not sure.  
 A: Let's say your variable is money spent on meat. In that case your spike at 0 are the vegetarians. You seem to be doing a regression model. Not taking the presence of vegetarians into account would be problematic. One possibility for that kind of data would be a Heckman selection model.
A: Adding to Maarten's answer, there are also the zero-inflated Poisson and zero-inflated negative binomial models. All these have been discussed here in the past. 
A: Observing a distribution that is actually the convolution of two or more distinct distributions is very common. 
In general, you can be observing the same phenomenon for a single population, but observe different causes leading to the need of multiple distributions to describe the data; or you might be observing two different phenomena in the same graph. 
As an example of the first, you might be measuring height for a male population with a ruler, and have Parkinson's disease; your distribution will then be the convolution of male's height, and your own tremor - both are Gaussians centered at the true value. Or you could be perfectly healthy and measure people's height using a sample of both males and females; you will then observe two partially overlapping distinct Gaussian distributions. 
Whatever is your case, there is no reason to ignore the subsample around zero, unless you think it's noise, and noise you don't want to model.
