Filesize limit on DOWNload?

I have a PHP script which copies files out of the client's directory tree and saves it to the local disk for further processing. This works fine except when the file is more than a megabyte in size. When it's "big" my resulting JSON has a string like this in it: The file "/home/username/public_html/wp-content/uploads/sucuri/sucuri-oldfailedlogins.php" for user "username" is larger then the maximum allowed file size of 1 MB."
When I went searching for solutions, I found pages on this forum talking about upload limits and how to fix them. Is there also a download limit and if so, how do I fix it? Moderator Note: Replaced the actual username with "username".