Cannot log in to Drupal site
The red box you see is a means that the e-mail address you typed in did not match with any e-mail addresses registered on the site, or that the password was not recognised.  You should also be able to see a red message-box   further up on the same page with more information about the problem.  For example:


  Sorry, unrecognized username or password. Have you forgotten your password?


The most likely explanation for this error is that you misremembered the password.  This also explains why it starts working again after you login as admin and reset the password for the non-admin user.

However, I've noticed that you've definitively ruled out this explanation in your question. Fair enough, I just wanted you to make sure this is not what is happening.  I've run dozen of Drupal sites for many years (many of them with the Email registration module enabled) - and I've never seen this happen except when the credentials are mistyped.

If that is not what is happening, I have no good alternative explanation.

To progress with debugging, you may extract the {users}.pass field (hashed) from your database, and see if the hashed value changes between when you're able to log in, to when you're not.  If it is changed, then you have some rogue process (or user) that munges the pass field in your {users} database.  (If it is not changed, then this is very, very mysterious.)