In the Mishneh Torah the Rambam states

Some of the Sages say that Elijah will come before the coming of the Messiah. But regarding all these matters and similar, no one knows how it will be until it will be. For these matters were unclear to the Prophets. Even the Sages themselves did not have a Tradition regarding these matters and only could attempt to understand the verses. Thus, there were disagreements in these matters. Nevertheless, neither the order that these events will occur nor their details are fundamental to the religion. Thus, a person must never busy himself with the Aggadohs and not dwell on the Midrashim regarding these matters or similar issues. He must not make them dogma. For these do not result in either love (for G-d) nor fear (of sin). Similarly, one should not calculate the Ends. Our Sages have said that the spirit of those who calculate the Ends will expire. Rather, one is to (simply) wait and believe in the principle of this matter, as we have explained. (Emphasis Mine)

which appears to discourage people to calculate the time of Moshiach's arrival. However in the Epistle To Yemen ,the Rambam gives a prediction himself. In addition, there are others who also give predictions as well. How is all of this justified?

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The Rambam himself addresses this in that very same epistle (speaking about R' Saadiah Gaon's prediction), after quoting the same Gemara he references in Mishneh Torah:

ואנו דנין את רב סעדיה לכף זכות ונאמר שמא מה שהביאו לעניין זה ואע"פ שהיה יודע שהתורה אסרה זה לפי שהיו בני דורו בעלי סברות רבות נשחתות וכמעט שתאבד תורת ה' לולי הוא ע"ה לפי שהוא גלה מן התורה מה שהיה נעלם וחזק ממנה מה שנדלדל והודיעו בלשונו ובכתבו וקולמוסו וראה בכלל מה שראה בדעתו לקבץ המון העם על דרך חשבון הקצים כדי לאמץ אותם ולהוסיף על תוחלתם והוא התכוון בכל מעשיו לשם שמים ואין לטעון עליו על שטעה בחשבונותיו כי כוונתו היה מה שאמרתי

As for R. Saadia's Messianic calculations, there are extenuating circumstances for them though he knew they were disallowed. For the Jews of his time were perplexed and misguided. The Divine religion might well nigh have disappeared had he not encouraged the pusillanimous, and diffused, disseminated and propagated by word of mouth and pen a knowledge of its underlying principles. He believed, in all earnestness, that by means of the Messianic calculations, he would inspire the masses with hope for the truth. Verily all his deeds were for the sake of heaven. Consequently, in view of the probity of his motives, which we have disclosed, one must not decry him for his Messianic computations.

So the same would apply to a lot of the other calculations, such as Rashi's.

The Rambam's own prediction, by the way, is not about the actual time of the arrival of Moshiach - he repeats, in the line before he gives his prediction, that this can't be known with precision - but about "the future restoration of prophecy in Israel." Which is, as he goes on to say, "one of the signs betokening the approach of the Messianic era" but not its actual beginning.

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    I appreciate the answer. I wasn't sure if the Rambam's reasoning for R. Saadiah was a blanket reason or there were different reasons for the others, which is why I asked the question. Jun 25, 2020 at 0:29

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