Depending on the nature of the chats, around a half dozen is my limit. If you've got 1-2 that are only responding to you every 10 minutes or so, or 1-2 have easy general questions, that number could probably push higher. But around 5 conversations that require legitimate investigation and troubleshooting is where I've found you can start to lose engagement.

I've found it's more about number of messages per minute, than number of chats at a time.

Where number of messages per minute is my real limit. (unless I am trying to trouble shoot a bunch of issues at the same time)

I find I can handle multiple chats at once easily using programs like TextExpander(Mac) or Breevy(Win).

That way, if someone needs a specific walkthrough, I can type something like oguide and it automatically pastes this link: https://www.olark.com/help/guide

Then if you model your support system around making the walkthroughs and guides for the most common user issues, it makes juggling chats much easier.

I've handle roughly six or seven at a time. Don't think I've hit eight. It can definitely get challenging with that many chats running.