Reddit user vsTerminus has discovered why CyanogenMod 11 and 12 are incapable of receiving MMS through Hangouts. He or she added that this works fine in the default Messenger app but does not if Hangouts is the default SMS application.
vsTerminus used the Shark for Root (Native) app to capture TCP messages while receiving MMS.
The user goes on to explain that a properly received MMS looks like this. Normally, the device downloads the attachment from the URL given by the MMS provider. Next, Rendering Configuration Information is sent from the MMS provider to the phone. The phone tells the provided that it got the file, and the provider deletes its own file. Lastly, an HTTP OK is sent to the device.
If this fails, then this happens. On the last step, the provider sends back an HTTP 204 No Content message. The phone believes the process failed when it already has the attachment.
vsTerminus determined that the MMS provided is probably not aware of the “CyanogenMod” user-agent given by Hangouts, and so it returns an error. His or her proposed solution is to switch back to Messenger as the default app or use a third party SMS app until it is hopefully fixed.