Is $a^{n+m}b^{n}c^{m}$ context free?
the corresponding grammar you gave would accept aaaabcbc. A better grammar for this problem would be

$X \rightarrow aXc$

$X \rightarrow aYb$

$X \rightarrow \lambda$

$Y \rightarrow aYb$

$Y \rightarrow \lambda$

where $\lambda$ is empty string.