Well, it turns out that it was DNS and apache related. Our systems administrators changed our DNS server a few weeks ago, and updated the /etc/resolv.conf file accordingly. They did not, however, stop and start apache (which was caching the DNS server value), causing gethostbyname() to fail. One of those cases where a restart does not cut it. DNS seemed like a likely source of the problem, as merlyn pointed out, but the fact that it was limited to apache was throwing me. Anyway, many thanks to all (especially tachyon) who helped me troubleshoot this (apparently OT) issue.
in reply to LWP::UserAgent connection problem as CGI