Too many files in one folder?

When I attempt to download backups of my sites, whether it’s through cPanel or my FTP program, the directories with more than a couple thousand files do not fully download.

Is there a limit on how many files I should have in a single folder on my server?

On many servers the LimitRecursion is set to 2000.
That means that maximum 2000 entries will be listed in your FTP-client.
Increasing the limit to e.g. 10000 may cause an overload, and in many cases ftp hangs.
You should consider putting files in separate directories.
E.g. put all files starting with character ‘a’ in a directory ‘a’, and so on.

hexburner is correct, it’s a setting in the FTP server, which most hosting providers set at 2000 (the default). You can bypass this limit by using SFTP (Secure FTP), as that is an SSH sub-system, which uses a different daemon; so it does not have that limit.