I can tell you first hand – that the problem is in fact both. With the change of WP versions, updating, etc. as WELL as server settings, specifically PHP that you need to modify, or have your host modify your php settings.
Customers that are hosted on cloudlinux operating systems generally have the ability to manipulate the versions of php (from 4.4 to 5.6, if their host has installed them all) and that includes settings, upload size, memory limit, path to the tmp folders, you name it – its there. Its easy to do so and it saves time. Especially if you’re coming from another host that used a different version of PHP.
On all my servers, I run the cloulinux with the php selector – and both on my end as the systems admin and the end user, we are all happy. When something comes up – there’s no editing any .htaccess file or php.ini file – this will take care of it, along with changing versions of php on the fly. Every single customer can run a different version of php and it has no effect on the others – making it basically flawless.
To that point – as I said, this problem stems from WP with their updates to the core, and it relies upon the server you are hosted upon. Don’t make the mistake of thinking its not both, because it likely is, or at least a good possibility, but the majority of the time, its a simple setting that can be changed within PHP.