I'm on a Slicehost 256MB VPS with Ubuntu 9.04 (Jaunty). lighttpd is the only web server process running; it listens on port 80. My lighttpd.conf can be found here. I'm using Ubuntu's default logrotate setup for lighty.
At seemingly random times, lighttpd will stop logging. It is not correlated with log rotation--that is, the errors do not occur when logrotate kicks in. What happens is, I will verify that the server is serving files by hitting a URL with my browser, and I will verify that it is not logging by checking access.log and seeing that the GET request I just made is not there.
Using init.d to restart the process starts logging again, without truncating or rotating the log file. That is, new requests will be logged at the end of the existing access.log file.
There are no cron jobs running on this box.