I have been taught that the last trumpet Paul says is blown at the Rapture, is the final trumpet of Rosh Hashanah. If this is true, then the Rapture can only happen at one time a year. Is this true or can the Rapture happen anytime?
You ask an interesting question. To answer it, let me begin by giving some background on the Jewish practice of blowing trumpets during their festivals. The type of trumpet used in Rosh Hashanah is a ram's horn (a shofar) that is blown during the feast, but the horn can also be called a trumpet. In fact, Jews commonly called the event the Feast of the Trumpets for that reason.
During the feast, the trumpet is blown a total of 100 times, with the final horn blast lasting much longer than the first 99 blasts. This final blast pictures the trumpet sound which will announce the Rapture of the Church, which Paul mentions in 1 Corinthians 15. By the way, the trumpet blast associated with the Rapture is not to be confused with the trumpet judgments of Tribulation. Rather, it is associated with the final horn of the Feast of Trumpets.
The feast of Rosh Hashanah is a shadow or picture of the Rapture, but we error if we try to mix too many elements of the picture God has given us with the reality it represents. Specifically, all seven of the Jewish feasts were given to the nation of Israel to picture various aspects of God's redemptive plan. These pictures taught men about the reality of God's plan of redemption and illustrated many if its details, but this does not mean that the festivals perfectly reflect every detail of God’s plan. Not every detail in the festivals find an obvious parallel in some historical reality.
For example, one day each year Jews celebrate the Day of Atonement, which is also a picture of Tribulation. However, the actual Tribulation we know will last seven years when it comes, but the Day of Atonement is only a single day on the calendar. The picture points to the reality but it does not substitute for the reality.
Obviously, it would be a mistake to assume that Tribulation will last only a single day simply because the feast that pictures the event lasts just one day. Likewise, it would be wrong to assume that the Rapture must occur on the very same calendar day as the Feast of Trumpets simple because the feast pictures the reality of the Rapture. Having said that, it is certainly possible that the Rapture will occur on the same calendar day as the Feast of Trumpets, but it is no more likely than any other day of the calendar.
Interestingly, in modern times the Jewish nation has taken to observing the Feast over a two day period rather than on a single day as originally given in the Bible. In centuries past, the Jews could not always be sure when one day ended and another began, since it was based on the visibility of three stars by eyewitnesses in Judea. Jews living outside Palestine were required to estimate when the day began in Israel and not based on where they lived (e.g., Babylonia or the Disaspora).
In order to be sure that they did not observe the wrong day, the Jews made Rosh Hashanah a two-day festival, thus leaving room for error in determining the start date. Technically, in Judaism today it is known as Yom Arichtaor (the long day), because the two days are counted as one rather than as two singular days. Consequently, even if the day of Rosh Hashanah was the actual calendar day God has appointed for the Rapture, we still can't say for certain which specific day it will occur, since the feast is currently observed over two days.