We are not aware of any law or rule that would explicitly require you to switch customers to paper statements in this situation. However, it may be prudent to do so. Assuming that the customer has agreed to receive statements electronically (as required by the federal Electronic Signatures in Global and National Commerce (ESIGN) Act and the Illinois Financial Institutions Electronic Documents and Digital Signature Act), the customer also has the right to withdraw consent, after which you must begin sending the customer paper statements. 15 USC 7001(c)(1)(A)205 ILCS 705/10(c)(1)(A). One could view a bounced email notification as a customer’s constructive withdrawal of consent, which is why it may be prudent to start sending paper statements to such a customer.