It all depends on where the RNG is being used. A good post with links to good reads.
Encryption: IV Generation
Encryption is effected because the entire point of an Initialization Vector is to provide more randomness into the crypt context without rekeying.
Encryption: Key Generation
I feel like this should be obvious, if your key is predictable in any way then what's the point in encryption? Any public key cryptography used to protect the key is invalidated because now the key can be derived, and communications decrypted.
Public/Private Key Generation
Weak key generation for asymmetric cryptography is extremely bad. If you can derive a private key it can be used to impersonate people or servers. Signed data with that certificate is now distrusted. Any session keys exchanged with its public key to encrypt data can now be exposed.
These can be indirectly effected if you're using an HMAC, this goes back to bad key derivation. If the integrity key can be derived, then an attacker can insert data into a stream and it will be accepted on the other side as legit data.
Now performing attacks against these weaknesses aren't always the easiest, but they're all doable. It depends on the protocol that's being used, how much other data is needed, etc. In a protocol like SSL/TLS you need data from both sides, plus the key shared in order to derive all keys. I hope this answers your question, I'd be happy to answer in more detail if you'd like.
Examples of when RNGs went wrong.