Scheduled reminders for memberships not fired

See also http://forum.civicrm.org/index.php/topic,28794.0.html

When I set up scheduled reminders for membership expiration 1 month in advance or 0 days (same day), no mail is being sent or is an entry added in civicrm_action_log, on both Mac (test) and Linux (live) servers. Both databases have been set up independently, but fed with the same membership data, but also after adding some test membership records and additional reminders I don’t get any reminders. I use crontab with cli.php, but I also tried to execute it from the Scheduled Reminders page.

The job finishes always with so-called “success”:
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Date Job Name Command/Job Status/Additional Information
2013-05-20 16:30:02 Send Scheduled Reminders
0
Summary
Finished execution of Send Scheduled Reminders with result: Success (1)
Details

Parameters parsed (and passed to API method):
a:1:

{s:7:"version";i:3;}

Full message:
Finished execution of Send Scheduled Reminders with result: Success (1)
2013-05-20 16:30:02 Send Scheduled Reminders
0
Summary
Starting execution of Send Scheduled Reminders
Details

Parameters parsed (and passed to API method):
a:1:

{s:7:"version";i:3;}

Full message:
Starting execution of Send Scheduled Reminders

______________________________________________________

Is there any version of CRM/Core/BAO/ActionSchedule.php with more comments? Then I could more easily help in finding the problem.