Quite simply, a website address is still resolving to an old IP/web server, displaying the content of an old website.
The nameservers for the zone in my account are the zone shows as live, so I know that my DNS is the ultimate truth.
However, the only record I have points to the new website server.
Usually, I’d experience a similar problem due to ISP caching, but it’s occuring with almost all ISP’s i’ve tested (never known it to go on this long).
I even primarily use CloudFlare’s DNS resolver!
Because even the resolved records are proxied (we load balance the web server itself), I can only ever see CloudFlare’s IP address.