After more testing, transfers are still not working reliably. Seems to work maybe once out of every ten attempts.
I have updated to v4.2.1 and changed the Auto Start option to "Through CGI daemon" and that hasn't helped.
It might be a problem on the server I am using at VPS.net, but when I switch back to Apache, transfers work fine, which points to a LiteSpeed issue.
FTP transfers with BackupBuddy seemed to be more reliable, but transfers to Amazon S3 or Rackspace Files... are always unreliable.
Do you have any other suggestions that I could try?