Does the application run on the same host as the database server? If the application is on a different host, try connecting from a client using the application's connection string, e.g. "sqlcmd -S np:\\myhost\pipe\mssql$myinstance\sql\query -U myuser".
Try connecting on the database server using the connection string as well as a connection string without the computername, e.g. "sqlcmd -S np:\\.\pipe\mssql$myinstance\sql\query -U myuser".
If you can connect on the sql server host with . (dot) as the computername, it implies that the computername is not resolving from the remote host. Possibly the IP address, hostname, or FQDN has changed in some way.