Upgraded to 3.5.1 and posted a @all message to announce it. On the plus side, the message post completed quickly.
Unfortunately the log filled up with 183 of these errors (we have 260 active users):
[2016/12/22 09:17:24 AEDT] [EROR] Failed to send mention email successfully email=redacted@domain err=SendMail: utils.mail.connect_smtp.open_tls.app_error, 421 4.3.2 The maximum number of concurrent connections has exceeded a limit, closing transmission channel