Your entire digital life could be stolen from you if someone knows your name, address, DOB, and last 4 digits on your card. Attacker can go through e.g. the Gmail password recovery process very convincingly impersonating you.
Gmail will let someone who has that information take over your email account?
The name+address+DOB of many people are public. So anybody who gets a glimpse at someones "card" (I guess you mean credit card?) would be able to take over their gmail? That would surprise me.
