X and Y are dependent and follow poisson distribution, but X+Y is not follow poisson distribution. Why?

Question

Question: X follow poisson distribution, Y follow poisson distribution, Give me an example that if X and Y are dependent, X+Y does not follow poisson distribution.

My starting is Let X=Y. So X and Y are dependent. But how about next? How to show that the sum of dependent Poisson random variables need not be Poisson with the sum of the rates.

Answer 1

If $X=Y$, then $X+Y=2X$ only takes even numbers.

This contradicts to Poisson distribution takes every nonnegative integer with positive probability.