If you can persuade them that there is a good reason (theft of mail from a flatmate etc), then persuade them that the new address is legitimate (like a relative) and get the new addressee to turn up at a branch with I.D then yes. But it's a palaver...
Some banks and building societies may be reluctant to send statements to an address different from your own as current statements are often used nowadays by official bodies and suppliers, such as mobile phone companies, as proof of address.