To combat this our staff are working on a login queue, as you might have seen during D2R Items. This will keep the population at the appropriate level which we currently have in order to be able to identify areas of the system that are straining and fix it before it shuts the game to a complete stop.
Every time we address the issue, we'll be capable of raising the population caps. This login queue has already been implemented partially in the backend (right moment, it appears as if there's a failure to authenticate your client) and should be fully installed in the coming days on PC, with console to follow following.
Players also have the option of being rate limited, but only in cases where games are constructed, closed and then reconstructed over short durations, which is mostly instances when players are farming zones such as Shenk & Eldritch or Pindleskin.
"When this happens it will be that there's an issue connecting to game servers. It isn't an indicator that game servers are down in this specific instance, it's just a sign that you are being rated limited to help reduce the load temporarily on the database in the interest of keeping the game running," Blizzard advised.
It all sounds like an absolute nightmare, to be honest, and I am so sorry for the engineers who have what looks like months of retroactive fix in front of them. There's an internet school of thinking that says, well, Blizzard should have seen this coming and prepared for it.
But that's also fundamentally part of the risk that you're taking with remasters. These cheap diablo 2 resurrected items games were written back in an age where multiplayer and information services didn't have the same popularity or accessibility is available today. Moreover, it's difficult to know whether the old infrastructure is as robust as we'd like it to.