Sometimes, when sending mails with attachments /var/log/exim4/mainlog says:

TLS error on connection from (hostname) [ipaddress] (gnutls_handshake):\
  Error in the push function.

I did not get around to dig deeper here, but - note to self: find out if this is related to #482012 and if setting MAIN_TLS_TRY_VERIFY_HOSTS to an empty value really helps.