Apache :: Intermittent Unknown SSL Protocol Error In Connection

We are hosting a Git server (using Gitolite v3). Our users are interacting with git over an https connection. Here's the relevant versions of software:

Apache 2.4.6
OpenSSL 1.0.1e-fips
Red Hat Enterprise Linux Server release 6.5 (Santiago)
git version 1.8.5.3 (Server Side)
git version 1.9.2.msysgit.0 (Client Side)

Users sometimes encounter an SSL error message when attempting to run git commands that talk to the server (e.g. clone, push, pull). The error message is "Unknown SSL protocol error in connection to servername:443". Users can usually simply try the command again, and it will then work successfully.

I've managed to capture a bit more output by setting the GIT_CURL_VERBOSE flag, see output below.

Code:

* Couldn't find host servername in the _netrc file; using defaults
* Adding handle: conn: 0x2587c40
* Adding handle: send: 0
* Adding handle: recv: 0
* Curl_addHandleToPipeline: length: 1

[Code] .....