As Voscaia alluded to, we know what happens to failures in the Accepted rings and the Aes Sedai Testing ter'angreal.

Accepted Rings:
1) If the illusion is more powerful than the women can bear, and she stays past the time the arches appear again, she vanishes, and never comes out of the world created in the rings
2) If she refuses multiple times, I believe it is three, to take the test in the first place, for whatever reason, she is put out of the Tower

Aes Sedai Test:
Passing depends on keeping serene while reproducing a certain combination of flows, all while being attacked or having your concentration disturbed in some way, so
1) If the woman cannot stay serene, she is put out of the Tower; the Aes Sedai value self-discipline and projecting an aura of serenity above all else, so she inherently is not fit to be an Aes Sedai of the White Tower. The important distinction to make here is that while an Accepted may be trained and disciplined enough to take and pass the Test itself, it takes years more of experience within her Ajah to learn their secret weaves, code words, eyes and ears system and other pieces of knowledge that actually makes her an Aes Sedai - this is why Moiraine is considered little more than a Wilder by some Sisters; between being Raised from Novice to Accepted and to the ring so quickly and being out of the Tower for so long so soon after gaining the ring and shawl, she barely knew enough to be considered a full Aes Sedai in the first place.
2) If the challenge overcomes the women, since the wounds are real, it is possible she is killed, and her corpse is the only thing visible when the test ends.

The woman herself, who gave up everything to come to the Tower in the hopes of being an Aes Sedai in the first place, is very likely to slink away in embarassment and become a member of the Kin or just keep as low a profile as possible. Keeping in mind that most of these women would not be among the strongest possible channelers in the first place, their "super powers" would be quite limited - it is a lot easier to justify a Child of the Light hanging a woman who throws fireballs around versus an Aes Sedai, who has the full resources and protection of the White Tower at her back. Even worse would be to flaunt being trained in the Tower, and draw the attentions of people who are hostile to Aes Sedai to you.