I’ve run into this issue a couple times and had to set everything aside and figure it out. It didn’t take long. I use WordPress Duplicator often to, well, duplicate and move different WordPress sites from production, for example, to a local dev installation.
Well, recently on two sites running the WordPress Duplicator the Zip Archive failed to load. After some searching I ran across this post which pinpoints the problem to the PHP version you are using.
So I simply switched the PHP version from 5.3.27 to 5.5.6. The post above, however, suggests downgrading the version to 5.3.x is what works. That post, however, is almost a year old, so I imagine something change on the Media Temple end.