Showing that $\{ c^n a^m b^{n+m} : n+m \geq 6\}$ is not regular
If $L_6$ were regular, then so would the following language be:
$$ (L_6 \cap a^* b^*) \cup \{ \epsilon, ab, a^2b^2, a^3b^3, a^4b^4, a^5b^5 \}. $$
However, this language is none other than $L_5$.

You can also use the pumping lemma. Try to pump the word $c^qa^qb^{2q}$ for large enough $q$.