Actually, no, according to Sky and Telescope. Modern folklore calls the second full moon in a month a "blue moon" but that definition is less that 60 years old. The original definition was the third full moon in a season that had four. This older type of blue moon can only occur in the month before the equinoxes and solstices (February, May, August or November) and usually falls around the 20th day so it could never be the second full moon in a month (which must fall after the 29th).
The sky and telescope article tells specifically how the error occurred, which is a fascinating, but somewhat long and convoluted, story. Here is an excerpt that explains the older definition:
The almanac also follows certain rules laid down as part of the Gregorian calendar reform in 1582. The ecclesiastical vernal (spring) equinox always falls on March 21st, regardless of the position of the Sun. Lent begins on Ash Wednesday, 46 days before Easter, and must contain the Lenten Moon, considered to be the last full Moon of winter. The first full Moon of spring is called the Egg Moon (or Easter Moon, or Paschal Moon) and must fall within the week before Easter.
[...] Seasonal Moon names are assigned near the spring equinox in accordance with the ecclesiastical rules for determining the dates of Easter and Lent. [...] When a season contains four full Moons, the third is called a Blue Moon.
Why is the third full Moon identified as the extra one in a season with four? Because only then will the names of the other full Moons, such as the Moon Before Yule and the Moon After Yule, fall at the proper times relative to the solstices and equinoxes.
So the blue moon was added in years that had an extra full moon, in order to keep the calendar straight for religious purposes. Since people have largely stopped caring about such things, it is hardly surprising that the newer definition, which is, after all, easier to remember, has become so popular.