Why is encrypting with the same one-time-pad not good?
I have no way to learn (say) $m_1$ unless I know $m_2$. 


That is exactly the problem - if you re-use the same key, and someone has access to one message you encrypted in both plaintext and encrypted form, they can use that to find your key:
$$
(m_2 \oplus k) \oplus m_2 = k
$$

As an alternative scenario, if you use the same key over and over, the attackers may be able to guess just pieces of various encrypted message, and each successful guess reveals a piece of the key $k$, so that over time more and more of the key is revealed. 

This general strategy for breaking a cryptosystem is known as a known plaintext attack. Many systems, like AES and RSA, are believed to be secure against these attacks. But a one-time pad becomes completely insecure against them unless a new pad is used for every encryption, which is why they are named "one-time pads".