Can two siblings in one family marry two siblings of another family?
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Yes. Many Mishnayos in Yevamos are premised on that very situation. See for example chapter 3 mishna 1.
Some have a custom to allow such marriages, but to have the siblings live afterwards in different cities. (This is sort of a compromise between the two positions that Shalom mentioned.)
The Lubavitcher Rebbe zt"l writes in one of his letters (Igros Kodesh, vol. 3, p. 182) that his father R' Levi Yitzchak Schneersohn and his uncle R' Shmuel Schneersohn were advised by the then Lubavitcher Rebbe, R' Shalom Dovber Schneersohn, to do exactly that when shidduchim were proposed for them with two sisters.
(To clarify, this applies only when two brothers are marrying two sisters. The question mentioned "siblings," which could also be a case of a brother and sister marrying a sister and brother from another family. There is no objection to that, as far as I know, and thus no reason to have them live in different cities.)
The Sefer Chasidim says not to, but the Noda Bihudah observes that the Talmud says it's fine. (See Barry's answer.)
The Noda Bihudah concludes that the Sefer Chasidim must have been talking about some special case. (Marrying someone with the same name as your parent is a similar case -- Talmud says it's fine, Sefer Chasidim says it's a problem; there, the Noda Bihudah suggests that the Sefer Chasidim knew through divine inspiration that any such marriage among his offspring would not be a happy one.)
(The above heard on a yutorah shiur from Rabbi J. David Bleich shlit'a.)
Regarding the same-name-as-parent issue, someone asked R' Moshe Feinstein. He said it's the kind of thing that if it doesn't bother you, don't worry about it, and if you're asking me whether to worry about the Sefer Chasidim, that means it doesn't really bother you, so go for it.