I'm pretty sure I've read that 3-4 am is when the body's circadian rhythm resets for the next day.
People who work nights often take a break around then because you can get silly/giggly and are not effective until the effect has passed.
It could well be that much 3am weirdness derives from the body going through the switchover awake rather than asleep.
EDIT TO ADD:
This is based partly on personal experience.
I used to do 12 hour nights for long periods when I was an aircraft mechanic.
We actually had very high productivity, as we tended to get stuck in and work without distraction on a specific set of tasks for which we would have prepared resources the previous shift.
However, the experienced guys would tell you to take a break around 03:00 or 04:00, as you tended to get giggly.
Some people had a sit down, some a small kip, and my preference was a few rags rolled up with duct tape to make a wee ball and which a few of us would kick around the apron for 20 minutes, with much laughter and comedy misses.
The point was, when you needed to be accurate, focused and attentive to detail, trying to push through the 'bonk' as it was often referred to, was pointless. I remember distinctly, one experience where I was absolutely baffled by the direction switch on a ratchet spanner. I spent minutes being unable to set it correctly before putting on a bolt head that was out of eye line. I just couldn't visualise the direction of turning and required action on the bolt. I then became frustrated and then laughed until a colleague called me a gobshite and pointed out the time.
Management were fully au fait with the practice and endorsed it as a safety enhancing.
I could easily imagine that the discombobulation of the circadian switchover, combined with a sleep transition, would result in much confusion and misinterpretation, resulting in the odd experiences reported for this time.