Fitting the Poisson distribution to binary data

I'm studying several time series. The variables of interests are dummy variables which take value 1 when a certain event happened, 0 otherwise. I want to find the best distribution to describe these data. I'm using the function fitdistr() of R, which allows me to use as discrete distributions the Poisson and the negative binomial. I've found using a chi-square test that the Poisson describe really well these data (p-values all over 0.90, so I do not reject the hypothesis that the data are distributed following a Poisson).

My question is: can a Poisson distribution be considered a good model to describe binary data? The hypothesis testing suggests that it is, I want to know if there are theoretical drawbacks or statistical errors in such fitting. Thank you.

If you know the only allowed values are 0 and 1, really the only sensible approach is to use the Bernoulli distribution (equivalent to a binomial distribution with $$N=1$$). The Poisson does converge to Bernoulli as the mean approaches 0, but if the mean value of the sample is greater than a small value (say 0.1), the Poisson will predict values >1 that can't occur in your data. For example for a Poisson with mean=0.1, $$P(0)=0.905; P(1)=0.0905; P(2)=0.00452; P(3)=0.000151$$ etc., while the Bernoulli would have $$P(0)=0.9; P(1)=0.1$$.