The problem is the execution time, in other words, the time it takes to upload the file, even though you have it set to allow up to 60 megs, you may not be able to deliver those 60 megs in the time allowed.
With the set time limit:
<?php set_time_limit(0); ?>
that should over ride the max execution time setting, but you are still getting the same error and the same report of the execution time being set to 600. this would mean the host has a setting in place preventing you from over riding the execution time. you will need to work with them to have the execution time increased.