I manage 2 accounts, but the problem is with the default one. Messages older than 60 days disappear from Inbox folder. The other folders are OK, I can see there messages older than 2 months. I have changed nothing in this account. The issue occurred maybe 1 week ago. Do you have any idea why this happened? 

Sounds like Automatic Archiving.

Go to Menu > Tools > Settings > Mail > Automatic Archiving and check if this is enabled.

No, automatic archiving is disabled. 

If this is an IMAP account, are the emails still visible through the web interface?