There are a lot of things that could go wrong, and a lot of tools to help us find out why it has. If you're familiar with the toolbox, including tools such as tcpdump, strace, ltrace, and pmap, you should be able to resolve most Zabbix problems.
Some people claim that everything is a DNS problem. Often, they are right—if nothing else helps, check the DNS. Just in case. Remember that it's probably a good idea to use IP addresses instead of DNS names for hosts in Zabbix.
As it would be quite out of scope, we won't discuss general Linux or Unix debugging here. Of course, there's still a lot of Zabbix-specific things that could go wrong. You might want to check out the Zabbix troubleshooting page on the wiki: http://zabbix.org/wiki/Troubleshooting. If that doesn't help, make sure to check the community and commercial support options...