Only if you use a very, very shitty wallet. I remember reading
this case (although it was about "R values" instead of "nonce k").
The r values mentioned is actually related to the k nonce, as this topic. You get the r value by multiplying (aka. Elliptic Curve Multiplication) of the k value with the secp256k1 generator point. Hence, the repeated k led to the repeated r.
The Android wallet fiasco is another big incident. This wouldn't happen with well known and developed wallets.
There shouldn't be a security problem with address re-use on Bitcoin Core because it's not using a deterministic nonce inside the signatures. It's using a completely randomly-generated nonce for all address types.
But even the wallets that do use a deterministic nonce most likely use RFC6979, which is still quite hard to break.
Bitcoin Core uses deterministic nonce for all the signing as of 0.10.0.
Probably as difficult as having to break the nonce without any information. If its correctly implementing RFC6979, then k is generated deterministic by both your private key and your data, which means it doesn't depend on CSPRNG at all.