There is a question asked, to which I have heard two answers:
Chazal teach us that there are many times when Moshiach will not come, such as on Shabbos or Yom Tov (Eruvin 43b). So how can one expect that Moshiach will come at any time?
I have heard quoted, but do not know the exact citation of, a teshuva of the Nodeh B'Yehuda (R' Yechezkel Landau) who was asked this question, and he responded that indeed one must expect Moshiach to arrive even on Shabbos, and when he gets here, he'll explain why it all works out. The assumption being that yes, at every moment one must be expecting Moshiach to come. (I have heard the same point quoted from the Yismach Moshe)
R' Yaakov Weinberg understood, based on this question, that the "at every moment"1 of waiting for Moshiach coming is not modifying the "that he will come," but rather the "I shall wait for him." In other words, one must at all moments be eagerly looking forward to his arrival, although there may be intervals during which his arrival is unexpected. (Source for R' Weinberg - recorded shiurim)
[It should be noted that the actual language of the Rambam in introduction to Chelek (Kapach translation) is:
והיסוד השנים עשר ימות המשיח והוא להאמין ולאמת שיבא ואין לומר שנתאחר, אם יתמהמה חכה לו
The 12th principle is the days of Mashiach, and that is to believe and to authenticate that he will come and not to say that he has been delayed, "If he tarries wait for him"
and the Rambam actually makes no mention of "every day" or "any moment." Nonetheless, the question of when we are supposed to expect him to come is still a valid one]
1These words are not a quote of the Rambam, who says no such language, but just an elucidatory phrase borrowed from the "Ani Maamin"