Last year, Amex limited each user to being able to save just one of each Amex Offer deal per person/SSN. We explained then that they still allow us to multiply that a bit by saving two varieties of the same offer: often, the same offer shows up in cash version and a points version, e.g. get $10 back in one version, and get 1,000 Membership Rewards points in another version. If you were lucky to get one card with the cash offer and another card with the points offer, you could save both of those and take advantage of the deal twice.
Tip: Reader Parts Unknown tells us to ensure we first save the points offer and then the cash offer. Apparently, if you save the cash offer first it will cause the points offer to disappear whereas if you save the points offer first the cash offer remains to be saved.
A lot of us value the Membership Rewards points more and instinctively save the points offer before saving the cash offer. That might be why I never noticed this before.
Update: there are various conflicting reports in the comments regarding this. One theory put out by a reader is that using the old interface or keeping multiple tabs open with each offer will help you save both offers.