Using Statement-I alone: Statement-I tells us that $(2x + y)$ is an integer. This means that the sum of $2x$ and $y$ is an integer. However, this does not provide any direct information about whether $(x + y)$ is an integer. Hence, Statement-I alone is insufficient to answer the question.

Using Statement-II alone: Statement-II tells us that $(x + 2y)$ is an integer. This means that the sum of $x$ and $2y$ is an integer, but it does not directly provide information about $(x + y)$. Therefore, Statement-II alone is insufficient to answer the question.

Combining Statement-I and Statement-II: From Statement-I, we know $(2x + y)$ is an integer, and from Statement-II, we know $(x + 2y)$ is an integer. However, even when combining these two, we still cannot directly conclude whether $(x + y)$ is an integer, as there is no clear relationship that would lead to $(x + y)$ being an integer.

Thus, both statements together do not provide enough information to answer the question.