I would suggest you grab and install this PHP script: http://phpsysinfo.sourceforge.net/
-- safe_mode must be off in order for it to work.
This will answer any questions as to how much memory the Server has and few others as well.
And they were suppose to have fixed the problem when we began paying for extra bandwidth.
This problem is related to Server resources and has nothing to do with Data Transfer.
What is the problem? What could be causing this?
Lots of reasons actually; little memory, too many scripts (as a whole) running on the Server, the scripts being used (even by yourself) may be outdated and using more memory than upgraded coding (if a home spun script) or a newer version would. What version of Perl is being used and are your scripts coded to take advantage of that version?
Downtime would be extensive and lost sales could kill us.
Why would there be any downtime? Whilst propagation is going on you have access to your new Hosting account and can transfer/setup everything before hand. Surely the scripts can be adjusted to use an IP address along with a Domain name -- giving you time to troubleshoot any bugs on the new Server.