Exactly what RNG weaknesses endanger (EC)DSA?
It is well known and trivial to show that if the same random $k$ is used in two different signatures, then the secret key can be extracted. In particular, you have two signatures $(r,s_1)$ and $(r,s_2)$--the $r$ is the same since it is a deterministic function of $k$--where $s_1=k^{-1}\cdot(H(m_1)+x\cdot r) \bmod q$ and $s_2=k^{-1}\cdot(H(m_2)+x\cdot r) \bmod q$. Computing $d = \frac{s_1}{s_2} \bmod q$ one obtains $d = \frac{H(m_1) + r\cdot x}{H(m_2) + r\cdot x} \bmod q$, and so $(d-1)\cdot r \cdot x = H(m_1) - d \cdot H(m_2)$ yielding $x = \frac{H(m_1)-d\cdot H(m_2)}{(d-1)\cdot r} \bmod q$.

One could think that this type of thing only happens if the exact same $k$ is used twice. However, if two $k$'s are related in some way, then the same thing can happen. This was shown in an interesting way in the paper Pseudorandom Generation in Cryptographic Algorithms: The DSS Case by Bellare, Goldwasser and Micciancio.

The above paper assumes a strong relation between the nonces. However, the point is that if the same nonce reveals the key, and if knowledge of the nonce reveals the key (trivial to show), then biases in the nonce can reveal information about the key. One would then need more signatures, but this can be done. This is shown in GLV/GLS Decomposition, Power Analysis, and Attacks on ECDSA Signatures With Single-Bit Nonce Bias.