“The database name could not be accessed. In order to access this database, check your network connection and verify that your database server is available. It may be necessary to disable any firewall software on your computer or server.”
Act! Answer ID 19640 http://is.gd/rpqb
In this user’s case the SQL Server (ACT7) service keeps stopping or doesn’t start when the laptop is rebooted (we’re tracking which). So I wrote a batch file (of which I’ve only done a couple) that will start the service. For those of you who don’t know, a batch file is just a text file saved with the .bat extention instead of the .txt extention and the contents contain a command for Windows to perform (careful that you have “Hide extensions for known file types” unselected in “Folder Options” -> “View”. In this case the command in the batch file is
::Turns on Act SQL Server Service
@ECHO OFF
NET START (or STOP) “exact name of service”
Programming is fun!! BTW, I stole this batch file from http://www.codingforums.com/archive/index.php/t-137177.html
"The question was, 'Have you rebooted?' not 'Do you believe that rebooting will help your situation?'" 
the answer ACT! won’t give you, but I believe is the best answer is as follows.
1. Do some research and find a different contact management program
2. have your database converted to the new program
There you have it. After a few weeks of learning the new system you will never look back and most likely not spend hours to fix “issues” that are in whole created by ACT! itself, not any other program.