The problem was that it was not possible to login to a WordPress website, to the admin dashboard I mean.

I tried many things:

• used scripts to reset the password. They actually change the pwd in the wp_user table. See below an example of update command used to change pwd.
• deleted cache plugins after having zipped them
• deleted cache folder, /wp-content/cache
• deactivated wp super cache plugin, which I had deleted before, by updating the option active_plugins in the table wp_options
• deleted anything related to caches in wp-config.php
• disabled all plugins by renaming the folder /wp-content/plugins
• enabled debugging

What worked was an upgrade from WordPress 3.9.1 to 4.4.2.

I used these instructions: https://codex.wordpress.org/Upgrading_WordPress_-_Extended_Instructions

1. USE name_of_your_database

2. UPDATE `wp_users` SET `user_pass`= MD5('yourpassword') WHERE `user_login`='yourusername';