A security "expert" advises: split bank data into groups, encrypt each group with a different algorithm and key, then compress the encrypted groups and encrypt everything again with yet another algorithm and key. What is wrong with this advice?

It adds complexity without security: grouping and cascades are unnecessary with AES, compressing ciphertext saves no space and adds no security, and key management gets much harder.

1. Grouping with different keys buys nothing. One strong key and algorithm protect all the data equally well; more keys only mean more keys to protect.
2. Compressing ciphertext is pointless. Good ciphertext looks random, and random data does not compress, so there is neither a space saving nor a security gain. Compression only helps before encryption.
3. Cascades are not needed with AES. A single AES-128 or AES-256 encryption is already beyond brute force. And as double DES shows, a cascade of two ciphers gives far less than the sum of the key lengths because of meet-in-the-middle.
4. Key management gets harder. Several algorithms and keys must be generated, stored, rotated and backed up, and every extra key is another place to lose data or leak secrets.

The real priorities instead: a single standard algorithm in a good mode, authenticated encryption for integrity, sound key management and secure implementation.