Tolkien's writing does not provide information on more than the first few centuries of this age, so it is not known when it ended, if it ever did.
Tolkien said that he thought the distance between the end of the Third Age and the 20th century C.E. was about 6000 years, and that in 1958 it should have been around the end of the Fifth Age if the Fourth and Fifth Ages were about the same length as the Second and Third Ages. He said, however, in a letter written in 1958 that he believed the Ages had quickened and that it was about the end of the Sixth Age/beginning of the Seventh. (This letter is referred to in the "Fourth Age" article of The Encyclopedia of Arda as mentioned below.)
The following is mere speculation and unrelated to fantasy works: That implies that maybe the fall of Nazi Germany ended the Sixth Age in 1945, as the other ages also ended with the downfalls of tyrants. The estimation of 6000 years would place the end of the Third Age and the start of the Fourth in the 5th millennium BC. Since Tolkien was a devout Catholic, it is speculated that he would have held the birth of Jesus to be the most important event of the era, and thus would be the most likely time for the start of the Fifth Age.
Determining the epoch of a Fifth era is important for those who apply the Tolkien calendar to present dates.
For a complete list of events during the Fourth Age see the Timeline