So in TDR the wondergirls find out that the Black Ajah had planned to get Taim freed after he had been captured.  Taim's escape happens but it never seems like the wondergirls or anyone else puts 2 and 2 together and figure that if the BA really did help him escape that they might not have turned him.  I just did a re-read of the series but didn't pick up on anyone pointing this out. 

There were past threads that discussed Taim's escape.

 

It seems unlikely that Black Ajah had any involvement in Taim's escape since Aes Sedia involvement in almost any known event seldom is kept quiet.

 

To me Taim's followers (from the area) helping him escape seems more likely since they would know the land better than an Aes Sedia would.

 

Joiya was the one who said that Taim was to be freed, and she was also not the one who was stilled.  The Black Ajah takes an Oath preventing betrayal, so if there was a Black Ajah plot to free Taim, and Joiya knew about it then she could not have spoken of it.

You have a point, but how did you know about the BAs oaths? (but it does make sense)

There is a point where Vandine and Adeleas are talking about interrogating the Black sister, I don't remember which where they suspect it and later Alviarin thinks about a Black sister not being able to betray her.

ohhh.... that makes sense

Alviarin states it with more clarity about Talene--she thinks to herself about Talene betraying the Black to Yukiri and Doesine 'if Talene HAD managed to get around her oaths...'

 

That being said, if Joiya had hear early plans for the freeing, but that Liandrin had decided against it, would allow her to speak of it. Had someone else decided to employ elements of that plan it would not have violated Joiya's oath.