A lot sooner than that!
My copy of Jean Meeus's Astronomical Tables of the Sun, Moon and Planets published in 1995 by Willmann-Bell Inc lists the following years as the most recent ones that had no Full Moon in February. Because all the other months have 30 or 31 days, it is impossible for them not to have a Full Moon. For February, there was no Full Moon in 1809, 1847, 1866, 1885, 1915, 1934, 1961, 1999, 2018, 2037, 2067 and 2094. During the most recent years, the missing Full Moon happened on March 2, 1999 at 6:59 UT; March 2, 2018 at 0:52 UT and on March 2, 2037 at 0:29 UT. Now, you will note that the Full Moons in 2018 and 2037 happen very early on March 2 as seen from the Prime Meridian where Universal Time is the Local Civil Time. This means that if you travel westward into the Atlantic Ocean, these Full Moons actually happen on March 1.