Moshe Rabbenu instituted the requirement of reading the Torah every Monday, Thursday and Shabbat, and many years later, Ezra added a requirement to call three Aliyot to the Torah on Monday and Thursday, and seven on Shabbat. The reason why he instituted only three Aliyot on Monday and Thursday is because of "Bittul Melacha" – the concern of lost worktime. Since Monday and Thursday are workdays, Ezra did not wish to overburden the congregation with more than three Aliyot, which would take time away from work. (The three Aliyot correspond either to the three groups of Jews – Kohanim, Leviyim and Yisraelim – or to the three sections of the Tanach – Torah, Nebi’im and Ketubim.)
The Mordechi (Rav Mordechai Ben Hillel, Germany, 1250-1298) tells of a case where three grooms were present in a synagogue on a Thursday, all of whom had gotten married the previous day. One of the grooms was a Kohen, and the other two were Yisraelim. The congregation faced the dilemma of to which of these two grooms they should give the third Aliya. The Rabbi there, surprisingly, ruled that a fourth Aliya could be added so that both could receive Aliyot. He explained that the week after a wedding has the status of a Yom Tob for the groom, and therefore, since we call up four Aliyot on Yom Tob, we may call four Aliyot to accommodate newly-married grooms so they could receive Aliyot. The Mordechi cites this ruling from the Risba (Rav Yitzhak of Shantz, France, late 12th century).
The Rama (Rav Moshe Isserles of Cracow, 1525-1572) codifies this ruling in his glosses to the Shulhan Aruch (Orah Haim 135:1). He adds that even if two Ba’aleh Berit – two fathers making a Berit Mila that day – are present in the synagogue on Monday or Thursday, a fourth Aliya may be added so they can both receive Aliyot.
Many other Halachic authorities, however, as noted by the Aruch Ha’shulhan (Rav Yechiel Michel Epstein of Nevarduk, 1829-1908), disputed this ruling. For one thing, the consideration of "Bittul Melacha" – not to overburden the congregation – is relevant even when two grooms are present in the synagogue. Moreover, even if the day is treated as a Yom Tob for the grooms, this is not the case for everyone else in the synagogue, and there is thus no reason to allow additional Aliyot. Indeed, common practice does not follow the Mordechi’s position, and we do not exceed three Aliyot on Monday or Thursday. When two Hatanim (grooms) are present, it is customary to give one of them the third Aliya and the second groom some other honor, such as carrying the Sefer Torah from the ark.
If, for whatever reason, a congregation was unable to read the Torah on a Monday, Thursday or Shabbat morning, should they conduct the reading in the afternoon, during Minha?
The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) writes (135:1) that the Torah reading was instituted not specifically for Shaharit, but rather for the day – Monday, Thursday or Shabbat. Hence, if the reading could not be held during Shaharit, it may be done in the afternoon, during Minha. This means that the Torah is taken out during Minha, and three Aliyot are read (or seven, on Shabbat), just as it would in the morning during Shaharit. In his Sha’ar Siyun, the Mishna Berura notes that this was the ruling of the Dagul Me’rebaba (Rav Yehezkel Landau of Prague, 1713-1793).
The Hida (Rav Haim Yosef David Azulai, 1724-1806), in his Haim Sha’al, disagrees. He contends that the institution of Torah reading was enacted specifically for Shaharit, and therefore a missed reading cannot be made up at Minha or Arbit. Hence, if a congregation was unable to read the Torah on a Monday or Thursday morning, they cannot make up the reading. If they were unable to read the Torah on Shabbat morning, then they make up the reading the following Shabbat. This is the position accepted by Hacham Ovadia Yosef, in Yabia Omer (4:17).
Summary: A congregation may not exceed three Aliyot on a Monday or Thursday, even if two Hatanim are present in the synagogue. If a congregation was unable to read the Torah on Shabbat morning, they can make up the missed reading the following Shabbat; if they missed a reading on Monday or Thursday morning, it cannot be made up at a later time.