American Express accounts need to be added one at a time. Otherwise all cards subsequent to the first on will fail.
Go through the connect accounts process and select only one account for adding at a time when presented with a list of available accounts on Amex's site. Subsequently repeating this for the remaining account(s).
Bank accounts, American Express (Amex)
Relevant to users on:
Money Dashboard Neon
Not sure which App you are on?
If you are unsure if you are on Money Dashboard Classic or Money Dashboard Neon? You can find that information here.