Steps to reproduce the issue

Attempt to update a fresh Joomla 3.5.0 installation to 3.5.1 with allow_url_fopen = Off in /etc/php5/fpm/php.ini, through the Control Panel.

Expected result

Well, obviously this should (and does) fail because of allow_url_fopen = Off in /etc/php5/fpm/php.ini but I forgot about that at first, and Joomla didn't make it clear that that was the issue, it in fact made it even more confusing because it was logging a successful download of the zip update file.

Actual result

You get a popup window saying the file is corrupt/truncated, etc.
Additionally, in logs/joomla_update.php this was logged:
File Joomla_3.5.x_to_3.5.1-Stable-Patch_Package.zip successfully downloaded.
And in the tmp folder this zip file had a file size of 0 bytes.

System information (as much as possible)

Debian 8.3 64-bit, PHP 5.6.17-0+deb8u1

Additional comments

Setting allow_url_fopen = On of course fixed the issue and I could then update without a problem through the Control Panel.