Can moon set after being up when due north? (pseudo circumpolarity) I'm in the northern hemisphere, facing north, and notice that the moon 
is due north (azimuth 0) and is still above the horizon, albeit barely. 

Does this mean the moon is necessarily circumpolar (doesn't set), at 
least for the next few hours? 

As the moon moves away from due north, it's altitude/elevation 
normally increases. However, it's possible that the moon's declination 
is decreasing so rapidly that the moon's elevation actually decreases, 
and the moon sets. 

Is this possible? 

This question is tangentially related to my question "libnova odd behavior for 89.5 degrees north latitude" on Stack Overflow.

EDIT to answer questions: I am facing true north. I am referring to the Earth's true moon, but not necessarily in any given time frame. In other words, if something like this happens 10,000 years from now, that's fine.

EDIT: OK, I believe the moon CAN set even if it's up when due north. Example: 


You are at latitude 89.5N 
The moon's declination when due north is +0.5N. This means the 
moon grazes the horizon when due north. 
If the moon's declination were constant, it's highest elevation 
for the day would be 1 degree. 
Since the moon's declination can change more than one degree per 
day (average of 1.5 degrees/day), it's quite possible for the moon's 
declination to go below -0.5N, at which point it would have set (it 
will actually set before this point unless it happens to reach due 
south). 


In theory, this could happen with any object whose declination 
changes, provide you're close enough to the North Pole. 

I'll leave this update open for critiques. 

EDIT: Actually, this whole question is just asking whether the midnight sun/moon can set at any position other than due north, which is fairly obviously true. Reasoning: once the moon/sun finally sets not due north, it was obviously still up the last time it was due north.
Yes - here is a good explanation of the situation. Just as it's possible to have continuous daylight for days or weeks on end, you can have continuous moonlight too.

The Moon's orbit is tilted at (a maximum of) just over 5 degrees to the ecliptic, however this tilt has an oscillatory precession with a period of 18.6 degrees. This precession means that our "lunar arctic circle" will vary in latitude over time.

For example, if the moon is sitting exactly on the ecliptic (latitudinally speaking), it would remain above the horizon until it is at a sufficiently low phase of illumination that its current ecliptic longitude is below the horizon. Of course in reality the moon only crosses the ecliptic twice during each lunar cycle (it's these two points that precess). Stellarium is a cool program to visualize this on if you turn on the ecliptic grids.

One important thing to note is that this phenomenon is dependent on both the time of year and the phase of the moon. In the summer months, new and near-new moons are the ones that can remain visible for days or weeks on end. Conversely, at the North pole, you might not see a full moon at all for a few months!