I have magento installed in a subdirectory. www.domain.com/subdir/magento
This site worked perfectly at one point. I changed nothing, until my client said he could
If a login error message ("invalid password", etc.) isn't being displayed it's almost always a session cookie problem. In order to rule it our entirely, use Use your browser's cookie viewer and/or your favorite HTTP traffic sniffer and check
That all the cookies have proper expiration dates after being set
That the session cookie has a consistent token name/value for each request
That PHP, when running through Magento, has the various session lifetime ini settings at a reasonable value
That PHP can write to whatever it's using as a cookie storage medium
That the server's time matches the real time, and that PHP itself has a timezone set