This article applies to the deprecated Zotero for Firefox (pre-Zotero 5.0) plugin. It no longer applies to the current versions of Zotero.
"The add-on could not be downloaded because of a connection failure on www.zotero.org."
This message previously could appear when attempting to download Zotero Firefox extensions on systems or networks that intercepted secure (HTTPS) connections to zotero.org. We've since put a workaround in place that allows installations to take place over such connections.
Automatic updates will continue not to work over such connections, however. See Updates Not Detected for more details.