According to an emailed halacha I received,
Some have the custom to not say Tachnun until the 12th of Sivan, and in the Diaspora until the 13th of Sivan. Shulchan Aruch w/Mishnah Brurah 494:3, Piskei Tshuvos 494:9
If one does not say Tachanun until the 13th, and the month, as is the case when the first of Sivan is on a Sunday, has 3 other Shabbatot and the first day of Rosh Chodesh Tammuz, then there are 16 days on which this person does not say Tachanun.
Following the premise as discussed in this and this answer, that once one does not say Tachanun for over half the month, one does not say it for the rest of the month, would it be proper not to say Tachanun for the remainder of Sivan?