I forgot to say that the message I receive is about an incorrect password.

Are username and display name different or the same? If they differ, you need to keep in mind that you need to enter the unique username to log in and not the display name (which could be the same - "Joe" - for several users).

I am the administrator of the Elgg site. My question is about creating a test user to verify the registration process. I have had to disable user registration and process new users by invitation, registering them from the admin panel.

Try disabling any 3rd party plugins and see if the problem remains.  Then you can enabled them one at a time and test to find out which one may be causing it.

What is your question exactly? As you say, you disabled registration and create each new account manually I doubt you ask about how to create a new user account as admin, or am I wrong?

If you want to create a test account in the way a new user would do it, i.e. via the registration page, you would need to enable the registration again. Then you only need an email address not used so far for an account to be able to create a new account the normal way.

Is the problem still the same as described in the first posting with login not working a second time? Have you tested if the problem with login is due to entering the display name instead of the username? If the problem is not due to the username I would follow the advice of Matt to disable 3rd party plugins to find out if any of them is interferring with the login.

The problem was about login a second time after the registration from the frontend. The problem is solved now. I believe, but I am not sure, that I haven't configured php.ini correctly and the user sessions table had almost as many entries as the log table. After configuring php.ini and restart the server the entries on user sessions table have decreased a lot and the login process works. Thanks.