How can I fix my server so core/plugins/theme update don't silently fail?

I run my own server and have a few dozen WordPress sites. Recently, many updates (plugin and core) have been silently failing. If I try to do the update from the updates page, I get a message like this one:

The update process is starting. This process may take a while on some hosts, so please be patient.

Enabling Maintenance mode…

Updating Plugin Jetpack by WordPress.com (1/1)

But it never progresses further. If I attempt the update from the plugins page, I get

Downloading update from http://downloads.wordpress.org/plugin/jetpack.2.3.1.zip…

Downloading update from http://downloads.wordpress.org/plugin/jetpack.2.3.1.zip…

But nothing more. There is nothing in either the PHP nor apache error logs, even if I set WP_DEBUG to true.

To be clear, this is happening for several plugins on all sites on this server, so I presume there's a configuration issue or other problem. But I don't know where to start looking to fix and/or debug the problem. Can anyone advise?

PS - The server is running WHM/CPanel, so it's CENTOS 5.8. Apache 2.2, PHP 5.3 (the problem also occurred in 5.2). The problem began when the managed support made some changes that were designed to stop the server overloading.