The specified network name is no longer available
Sounds like spotty DNS resolution issues or long running queries.
For DNS trouble-shooting
As a Band-Aid and trouble-shooting aid add and entry to the hosts file on your app servers (not the SQL Server) at c:\windows\system32\drivers\etc
SQLServerIPAddress SQLServerName

Example:
172.16.0.5 ProductionSQLBox

That way, the SQL Server name will be resolved by the hosts file until you can find the real problem for what's going on with name resolution.
Be sure and test the hosts file by pinging the SQL box from the app server via a command line. Or, alternately create a DSN with the ODBC Administrator within Control Panel and test the connection there.
For long running queries
Implement a longer command timeout value for the application connection strings connecting to the SQL Server.
Example
&lt;connectionStrings&gt; 
    &lt;add name=&quot;webconfigconnectionString&quot; connectionString=&quot;server=SQLServerName;database=dbName;uid=u‌​serName;password=ABC‌​123;Connect Timeout=120&quot; /&gt;
&lt;/connectionStrings&gt;