Yes, but only when the month is February.
However, the length of an actual lunar month can vary from this average figure by as much as seven hours, because the Moon's orbit around the Earth is not circular, and neither is the Earth's orbit around the Sun.
As a result of this non-circularity, the Sun and Moon move at varying speeds along the ecliptic. Sometimes, the Moon is moving faster than its average speed and the Sun is moving more slowly, so the Moon "catches up" with the Sun sooner, making a shorter lunar month.
At other times, the Moon is moving more slowly than average and the Sun is moving more quickly, so the Moon "catches up" with the Sun later, making a longer lunar month.
You can view a diagram (as a PDF file) which shows the variation in the length of the lunar month during the period 1980 to 2020.
Notice that the longest lunar month is four hours shorter than 30 days, so even a 30-day month such as April or June must contain at least one Full Moon.
The only month which could miss a Full Moon completely is February. This is most likely to happen in a non-leap year, when February is only 28 days long, but even when it has 29 days, this is still less than the length of the shortest lunar month, so it's possible for any February to miss a Full Moon.
Since a thousand years must include a thousand Februaries, it is obvious straight away that 48 of those Februaries are missing a Full Moon.
Counting the number of Full Moons which fall in February in a leap year, we find 240 of them.
However, the thousand-year period from 2000 to 2999 has 243 leap years. (Remember that in the Gregorian calendar, century years are only leap years if they divide by 400, so 2100 will not be a leap year, and nor will 2200, 2300, 2500, 2600, 2700 and 2900.)
So there are 243 leap years, but only 240 of them have a Full Moon in February. This means that there will be just three leap years in which February will have no Full Moon.
Those years are 2572, 2792 and 2944.
It turns out that in those three years, both January and March have two Full Moons, but that isn't surprising since a 29-day February can only miss out on a Full Moon if there is a Full Moon late on January 31st. That, in turn, means that there must have been a Full Moon in early January, and it also means that the next Full Moon must be early on the morning of March 1st, thus March will also have a Full Moon at the end of the month.