Since a full moon happens once every 29 days (and that's not exact, but close enough), the longest period of time where you can have 11 full moons but not 12 is 11 x 29 + 28, or 347 days, so in a normal year, it's not possible. There are slight variations in the time between full moons but not nearly enough to make up the extra 18 days needed.
So you'd need a special circumstance. Great Brittan's short year in 1752 wasn't sufficiently short, it had 11 days removed.
There is a phenomenon where not everywhere on Earth gets a full moon every 29 days. In the North and South poles for example, where during summer they get 24 hours of sun and winter, no sun, a somewhat similar effect happens with the moon. The Moon orbits the Earth at 5 degrees off the ecliptic or basically, the plane between the Earth and the Sun and that makes the Moon's apparent movement across the sky a little more complicated than the Sun's, but because full moons only happen when the moon is roughly opposite the sun, it's not hard to see that when the sun is in the sky for 24 hours during the Polar summer, they don't get a full moon for several months because the full moon is below the horizon when the sun is in the sky 24 hours. You might only see 6-9 full moons a year at the poles (if I was to guess), and there's probably a range of latitudes close to the poles where 11 full moons might be about the yearly average.
If you want to split hairs even further, the Moon only takes an instant to pass the 180 degree point in relation to the Earth and Sun, so in a sense, a perfect 180 degree full moon is only visible to half the earth and by the time the other half of the earth turns to make that moon visible, the moon has already gone past 180 degrees and from a certain perspective, it's no longer full. But, a full moon is generally regarded as a roughly 24 hour period, not an instantaneous moment, so I don't like that argument personally.
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