In the case of a screen saver, it is possible (though uncommon) that an application may delay, prohibit, or otherwise interfere with other background processes, such as the process that allows the workstation to respond to this status request. Some applications preempt these background processes to allow the application to perform optimally for the user, though in such cases they will USUALLY provide user-definable options to "allow background processes" or "monitor system activity," and so forth. Check with the software manufacturer to confirm this possibility. Finally, it could be that your screen saver allows background processes, but conflicts with the system or network software.
As to being unavailable, very 30 seconds, the computer running the Network Assistant application will poll the network to verify the availability of workstations configured under Network Assistant. Any workstations that fail to answer this "are you there" request will be marked unavailable. As the request is continually delivered, correcting for a workstation's inability to respond will automatically make it available again after a subsequent status request is answered.
Typically, failure to answer these status requests can be attributed to common network connectivity issues: a break in the cabling or connections, a powered-down computer, a bad network card or transceiver, and so forth.