CBC is okay if it is used to encrypt before MACing. The issue is that TextShredder used CBC without any authentication, i.e. a MAC. That means someone can modify the ciphertext in transit and it won't be detected. This opens up several types of attacks. In theoretical terms, unauthenticated block ciphers are not secure against adaptive chosen ciphertext attacks (CCA2). In practical terms, one attack that comes to mind is a padding oracle attack. Essentially, an attacker given a real message and can create modified ciphertexts that will cause a decrypting party to potentially leak information in a side channel.
Serge Vaudenay talked about this 12 years ago:
http://www.iacr.org/cryptodb/archive/2002/EUROCRYPT/2850/285...
The recent Lucky 13 attack was a form of this type of padding oracle attack:
https://www.imperialviolet.org/2013/02/04/luckythirteen.html