Yoshi-P said they were at "maximum concurrent connections of over 230,000 users" most recently, and with 50 servers that comes to more than 4600 per server on average. The limit was less than that for a while, until SE optimized the duty finder and stuff.

There seems to be a queuing system on each server, and its capacity seems to be a few hundred, though they might have scaled it back for stability. I've been at #433 early on, but more recently I've been at double digits. When that queue fills up, I think that's when we get the 1017.

How many total per server? SE hasn't said. I would make wild guesses, but they might get quoted. So between "a lot more than 5000" and "a whole lot more than 5000".