You can telnet to an application port to test connectivity. For example if your database is listening on port 1433, at a command prompt, type telnet host.ip 1433 or change the host.ip with the hostname to test name resolution. You can do the same test if you are using FTP services or simply use an FTP client (e.g. filezilla). If you permit ICMP in your network, simple ping testing will tell if the path is working between client and host. These are for testing layer 3/4 and below.
For testing above layer 3/4, you will need some application specific tools for your environment and access to the system logs. Turn on auditing during your testing to see if the events are being properly captured.