Dov Bear has a post on gebrochts, the custom of not eating matza that has in any way become wet. Dov Bear's point, based in part on Rabbi Eliyahu Kitov (a Chabadnik), is that the custom has no halakhic source. If memory serves me, the Rema* mentions the custom and is also puzzled by its source, and posits that children saw their parents not eating wet matza on the first night of Passover and confused this custom, creating the new "minhag" of gebrochts. Why not eat wet matza on the first night of Pesach? An opinion of the Rambam (here brought by Amshinover in the comments to Dov Bear's post):
[W]e quote from Nefesh HaRav by R. Hershel Schachter, shlita, Rosh Kollel of R.I.E.T.S. This sefer represents the views of his rebbe, Rabbi Joseph B. Soloveitchik, zt"l. R. Schachter writes (p. 18: "Even though mitnagdim [as opposed to Chassidim] are not accustomed to refrain from eating gebrockts (matza sheruya) on Passover, the Beit Halevi [the Gaon R. Yitzhak Zeev Soloveichik] and after him his son, the Gaon R. Chaim Soloveichik, were both careful to refrain from eating gebrockts on the first day of Passover because of Rambam`s ruling [in Hilchot Chametz U`Matza] that one should not eat matza ashira on the first day of Passover, even when not eating it for the specific purpose of mitzvat matza at the Seder. Cooked matza is likened to matza ashira in this regard, and this chumra (stringency) is a tradition from R. Chaim of Volozhin."
Matza ashira (grape matza, egg matza, etc.) is extended – without precedent in any halakhic literature or practice – to include shemurah matza broken up into soup. This morphs into never eating any wet matza. Which morphs into the Chabad custom of not eating matza that has come into contact with any food. All of this because of the fear that unbaked flour within the matza will somehow rise when brought in contact with liquid (or, in the case of Chabad, solid as well).
But to really understand this custom, one needs to look back to its origin and then slightly before that. The custom is not much older than 500 years old, if that. And it began not long after another Passover custom got its start – thin, cracker-like matza. Before this time (and in many Sefardic communities to this day) matza was soft and thick, like a large Arabic flat bread. These matzot were sometimes several inches thick, but more commonly 1/2 to 1 inch in thickness. These were the standard matzot eaten by Jews the world over until a custom began in (where else) Eastern Europe among those identified with Ashkenazim – for fear of hametz being found in soft matzot. This probably took place after these communities first came in contact with Europeans who baked water crackers and saw what they felt was an advantage in the process. Soon matza was wafer thin and baked at extremely high heat. And our flat matza was born.
But all was not rosy in matza land. The rapid high temperature baking of the new matza often left pockets of under-baked dough within the matzot. Although halakhicly these matzot could not become hametz, the especially pious (and the ignorant) nonetheless feared the possibility. I believe it was then that the "custom" of gebrochts entered the Jewish community, and the memories of zeiydi who didn't eat gebrochts on the first night of Passover because of a stringent reading of the Rambam blurred into zeiydi who didn't eat gebrochts, period, because of fear of hametz.
There is no halakhic source for the custom. It is a custom described in halakhic literature as foolish. It was propagated largely by the then-nascent hasidic movement, whose hallmark was anti-intellectualism and anti-rationalism. And now, the same people who rioted in Boro Park and Jerusalem are sitting around their Passover tables believing they are frummer than everyone else, in part because their matzot don't see the insides of soup bowls. And, perhaps even more absurd, thousand of Chabadniks all over the world eat their burned shemurah matza from plastic bags to prevent matza crumbs from contaminating their tables and tableware, just as Moses did in the Sinai desert 3300 years ago.
* Can anyone cite this souce?