On re-reading I realized my previous post might not have been really clear, so here goes in trying to clarify.
Let's say you go to /home/bigone and do a du, finding that you have 1Gb of files in public_html/bigfiles. I'd move /home/bigone/public_html/bigfiles somewhere else, then tar it up and send it over to the new server. Then I'd check the size of /home/bigone and if it was 500Mb or so, I could then go on and check the databases and do the same thing there. The idea being to reduce the size of what cpanel needs to transfer to be well under 1Gb so it goes faster and works (sometimes cpanel barfs on big transfers, even if they're under the maximum file size, best to keep things well under 1Gb and if that fails, try under 500mb).
It's still worthwhile using the cpanel transfer as it creates all the email accounts, forwards, ftp users, .htaccess files, etc etc, so is a real time saver there as that's the stuff that takes ages to transfer.
Cooldude - I guess that Apache limitation may be why the operation is failing, although I'm not sure it's actually through Apache at all? Unless the account tarball is being downloaded via an HTTP GET?