After the mabul, Hakadosh Baruch Hu entered into a bris with mankind that He would never again destroy the entire world via a flood. The rainbow was chosen to symbolize this bris. This is why one who sees a rainbow recites a beracha asking Hashem to remember and uphold His bris.
The Talmud (Kesubos 77b) states that when R' Yehoshua Ben Levi died he was greeted in heaven by R' Shimon Bar Yochai who asked him whether a rainbow had ever appeared during his lifetime. When R' Yehoshua replied that it had, R' Shimon commented that apparently R' Yehoshua was not that great a tzadik. The rainbow would not have appeared in a generation of a very great tzadik, because in his merit the entire generation would have been spared, and thus there would have been no need to invoke the aforementioned bris and have its symbol appear.
The Talmud proceeds to comment that in truth the rainbow had not appeared during his lifetime. R' Yehoshua Ben Levi had lied to R' Shimon out of humility. If he would have told the truth it would have appeared as if he were bragging. The Talmud (Bava Metziah 23b) tells us that a talmid chacham may lie in three types of situations, one of them being for the sake of humility, that others should not know how learned or how pious he is.
The Shulchan Aruch (Orach Chaim 565:6) applies this idea, stating that if one has accepted upon himself extra fast days and lets others know of his middas chassidus, "he will be punished for doing so"[1]. Every so often in Shulchan Aruch and in its commentaries we come upon a recommendation that a ba'al nefesh ought to be stringent on himself regarding some halachic issue. Rabbi Yehuda Amital shlit"a relates that when he was a teenager learning in yeshiva, when the students learned Mishna Berura and came across a statement that "one who is a ba'al nefesh should act on accordance with the stringent view", they thought to themselves that this certainly refers to a few outstanding tzadikim who they knew of. Whereas today, he further commented, many of the yeshiva students instinctively assume that it refers to them!
Even one who does belong in the category of ba'alei nefesh must be careful that his middas chasidus not turn into a public demonstration of his piety. Today the term "frum" is usually used with a positive connotation. In pre-World War II Litta it was usually used as negative description of one who publicly displays his piety. Some of the ba'alei mussar used to say "a galach has to be 'frum'; a yid has to be 'erlich'". They would add that the word "frum" is roshei teivos (an acronym) for "fiel rishis, veinig mitzvos – much evil and very few mitzvos".
Additionally, a middas chasidus is not to be forced upon others. The expression in the Shulchan Aruch always is that a "ba'al nefesh yachmir al atsmo – a ba'al nefesh ought to be stringent upon himself."
The Talmud (Berachos 35b) states, "Many attempted to follow the style of R' Shimon Bar Yochai and they did not succeed." The style of the ba'al nefesh, following the stringent approach, was clearly not intended for the masses. Those who are able to should certainly strive to attain the state of chassidus, but this must be done step by step, as spelled out in the Talmud (Avoda Zara 20) and elaborated in Mesilas Yesharim.
[1]The Taz and Magen Avraham quote the aforementioned passages (Kesubos 77b and Bava Metzia 32b) in their comments to this line Shulchan Aruch