Updates cause site to crash

I recently updated my plug ins and my site seemed fine initially. Then it began freezing. I contacted my hosting folks at WPEngine and they say this is at the head of my site and not resolving:

I restored my site to a date before the updates and all seems OK now.

Now, I decided to update again - and try to narrow down the issue. However, now I get this error when trying to update plugins. I'm using almost exclusively WPMUdev.org

Here is the error:

An error occurred while updating Affiliate: Could not copy file. /nas/wp/www/cluster-1186/websitewag/wp-content/ et. etc (different path for each plug in)

Any ideas?