Plugins no longer updating since installing a few big WPMU plugins

I have installed Hummingbird Pro and Snapshot Pro in the last week or two and have been configuring them (the documentation for both of these is quite outdated / non-matching throughout them). Somewhere along the way, all our plugins on the site have stopped updating. They return "Update Failed: Plugin update failed" messages every time. Of course I don't know that it's because of either of the two plugins I mentioned, but we haven't changed anything else major. I thought it was maybe because Snapshots had filled our hosting account up, but I've since emptied out most the backups and finally managed to figure out how to send snapshots to Google Drive (I think) and we're still having the problem. Besides, we're supposed to have "unlimited" storage on our hosting anyway. I've also checked our file permissions and all sorts of other things I Googled to try. All that to say, the plugins are still not updating and we can't figure out why. Any ideas as to why?

P.S. I have already enabled Access from a different support ticket asking for help checking out our Hummingbird settings. So, feel free to hop in and look around.