Can you get more than one signup bonus with two Amex card applications?

I have never held an AMEX card but am now interested in getting the Explorer and possibly the Charge.
If I apply for both at the same time, will I get bonus points for both since I don’t have any AMEX card? If not, which bonus point offer will I get?

you will only get bonus points on the first one you applied for. this should be plainly obvious.