For EC Pedersen commitment: The two generators are G and H.  Two messages and randomness are $m_1$, $m_2$, $r_1$, $r_2$,
so the two Pedersen commitments are $Gm_1+Hr_1$ and $Gm_2+Hr_2$.

When adding these two, I got a new Pedersen commitment as $G(m_1+m_2)+H(r_1+r_2)$ with message $m_1+m_2$ and randomness $r_1+r_2$. But then what if the message $m_1+m_2$(or randomness $r_1+r_2$) overflows?

For example messages are in field mod 2^64, than if message becomes some 2^64+1, it would become 1. As G*(2^64+1) should not equal to G*1, unless G has the order of 2^64.