We have a few anecdotal reports of bonus denials for people who did not see the Pop-Up. I doubt that the American Express system 100% reliable with every operating system, browser, security settings combination, but I am also confident that many people do not read carefully and might click thorough a warning without remembering.
I was just denied a 50K gold personal card bonus recently with no explanation. No pop-up or any other notice that I was ineligible for the bonus. Application was directly from the american express website with my user ID logged in and my browsing to the "explore cards" tab. I have never held a personal gold or platinum card. Still "fighting it" by leveraging biz centurion customer care; not looking hopeful.