This issue bugged me a lot. So, this is for the record:

Setup:

Firefox/Chrome/IE -> Apache Reverse Proxy -> Zimbra Server

Issue:

Firefox users saw a lot BAD_JSON_RESPONSE errors in some folders. Chrome users did see it a lot less. IE users had the problem too, but I cannot tell anything about how often the issue occured.

Cause:

According to the Jetty log, the Zimbra server send about 40kB of data. But the reverse proxy only send about ~20kB of data to the client later. So the JSON response was incomplete and not parsable. From the Apache error log:
(28)No space left on device: proxy: error reading response
Of course, there is plenty of harddisk space available.

Fix:

Upgrading the Apache fixed the issue. More detailed:
The Apache within Debian Stable (Squeeze, 2.2.16) has this issue while the Apache from Debian Testing (Wheezy, 2.2.22) does not have it.

Last words:

I could not identify an Apache bug report nor a specific changelog, that describes this behavior.