Which condition does not satisfy that E [X + Y] is equal to E [X] + E [Y]?

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Since E[X + Y] has linearity, it can be resolved to E[X] + E[Y]. However, it is said that this method does not always work.

E[X + Y] = E[X] + E[Y]

What are the conditions under which the above equation cannot be achieved? and why?