Let $N$ be the greatest common divisor of $N_1$ and $N_2$, is it true that there exists a congruence subgroup containing $\Gamma(N_1)$ and $\Gamma(N_2)$ but not containing $\Gamma(N)$, where $$\Gamma(N)=\ker(SL_2(\Bbb{Z}) \to SL_2(\Bbb{Z}/(N)))$$ is the full congruence subgroup of level $N$.
If $(N_1,N_2)=1$, then $\Gamma(N_1)$ and $\Gamma(N_2)$ can generate $SL_2(\mathbb{Z})=\Gamma(1)$, thus the proposition is not true in this case, but I do not know the general case.