The Shulhan Aruch (Orah Haim 141) writes that although from a strict halachic standpoint two brothers, or a father and son, may be called to the Torah for successive Aliyot, this is not done, due to the concern of Ayin Ha’ra (an evil eye). Even if the family members say they are not concerned about the dangers of Ayin Ha’ra, they should nevertheless not be called to the Torah for successive Aliyot.
The principle of "Beneh Banim Hareh Hem Ke’banim" establishes that the relationship between grandchildren and grandparents is halachically akin to the relationship between parents and children. As such, the custom is not to call a grandfather and grandson for successive Aliyot, just as a father and son are not called for successive Aliyot.
Two brothers should not be called for successive Aliyot even if they are only half-brothers, who share just one parent.
Two family members may be called for Aliyot that are not successive; as long as there is at least one Aliya in between, they may be called for two Aliyot.
Moreover, two family members may be called for successive Aliyot that are read from two different Sefer Torah. For example, on a Shabbat when two Sifreh Torah are used, such as Shabbat Rosh Hodesh or Shabbat Hanukah, two family members may receive Shebi’i and Maftir, since these Aliyot are read from different Sifreh Torah. This is the view of numerous Halachic authorities (including the Shiyureh Kenesset Ha’gedola, Elya Rabba, Havot Yair, Hida in Le’david Emet, and Mishna Berura 141:20.) This Halacha applies even when Kaddish is not recited in between the two Aliyot that are read from different Sifreh Torah, such as in the case of Shelishi and Rebi’i on Hol Ha’mo’ed Pesach, which are read from two Sifreh Torah without Kaddish in between. Another example is Hatan Torah and Hatan Bereshit on Simhat Torah, which are likewise read from different Sifreh Torah without Kaddish in between.
In situations of need, a son-in-law and father-in-law may be called for successive Aliyot. This is the ruling of the Petah Ha’debir (Rav Haim Binyamin Pontrimoli, Turkey, d. 1873). Rav Haim Palagi (Turkey, 1788-1869) writes that although some authorities dispute this ruling, there is room to be lenient in this regard in situations of need. Hacham David Yosef, in Halacha Berura (p. 214), writes that he personally witnessed his father, Hacham Ovadia Yosef, permit calling up a son-in-law and father-in-law for successive Aliyot.
Summary: Two family members – father and son, two brothers (including half-brothers), or grandfather and grandson – may not be called for successive Aliyot, unless the two Aliyot are read from different Sifreh Torah. When the need arises, a father-in-law and son-in-law may be called for successive Aliyot even if they are read from the same Torah.