I am not sure if there is a hard-cap limit, and if it does it would depend on the browser, I don't think clients should be limiting connections when comes to publishing notes, at least I don't do it on my client, nor limit the relays for the current logged user when fetching his own info.

When comes to main following feed, I do a 3 max relays per person per subscription which is configurable, my feed is usually around 100 relays connected out of around 240 relays attempted, sometimes I logged as read-only with Derek npub and set to 14 relays per person to see things fire, it tries to connect to 780 relays but only around 160 was connected.

I got into situations of opening multiple tabs and some tab couldn’t opened single connection, also other issues of "insufficient resources", but those doesn’t happened always.

I was working on a better dev-tools for my client to better visualize this but it’s not ready yet.