I recently had a client where everything was working fine - but they were unable to link to any secure (https) sites when creating entries in Movable Type. Unfortunately, there were no messages to be found, and everything worked just fine - as long as there were no https links in the entry.
To make matters worse, just changing the https link to http made it work just fine - which at first seemed to make no sense whatsoever. It's like that one character put the database over some sort of size limit or something. Alas, even that wasn't it - bunches of (other) characters would work fine, just not those particular characters, making it seem like that site itself was the issue, and that is what eventually led to the answer.
Many sites I work on these days have turned off TrackBacks, and in a related vein, have also turned off outgoing pings. But this one had not - so when I started following the process through, I noticed that it was taking longer than it should have to save the entry.
It seemed that it was trying to parse the various links in the entry in order to send outgoing pings, and this is where the problem was. Unfortunately I was not able to determine if the problem was with the site itself or the SSL protocol, but I was able to divine that turning off outgoing pings, which really aren't needed by much of anyone these days, took care of the problem.
After disabling the pings, and re-saving the entry (with the https link intact), it worked completely as intended. Problem solved!