Oh, sorry. Seems the reason is different - it's just "Test Connection" button in client. It gives an auth error message anytime the password is hidden behind asterisks. If I just paste plain text in password field instead of asterisks, test gives me successful connection (even without saving).
So here's my client version (installed from amd64.deb from http://mattermost.com/downloads/):
Mattermost Desktop 1.2.1
And my system:
$ uname -a
Linux comp 3.16.0-4-amd64 #1 SMP Debian 3.16.7-ckt25-2+deb8u2 (2016-06-25) x86_64 GNU/Linux
$ cat /etc/issue
LMDE 2 Betsy \n \l
Just checked from browser - it's the same error.