Okay, there are several theories surrounding the identity and past of Tam al'Thor. I have a few questions and/or comments about these theories.
1) It's been said that Tam is Jain Farstrider. Well, since we have no true proof that Jain is dead (I think), how old would he be if he were still alive? I think that will tell us whether or not he could be Jain.

2) I highly doubt that he was ever a Warder. If Kari were AS and he moved away with her, the Tower would most likely know and would have been able to track them down. Also, he served in the Illian Companions. Warders usually serve the White Tower only.

3) The theory I think is closest to the truth is that Tam is ta'veren. I think maybe he was under ta'veren influence rather than actually being ta'veren himself. It was mentioned that farmer ta'veren can become kings and Tam's rise from farmer to Second Commander was used in comparison. What if the Dragon was influencing his destiny from Tel'aran'rhiod (where the Heroes of the Horn end up) so that he would become the Second Commander and end up in the right place to pick up the baby that would benamed Rand?

This has nothing to do with my actual theory, but it's been said that he locked the door and got his sword because he sensed Trollocs (due to being a Warder in the past). I think the bond is what gives a Warder his powers, and without it, he loses them. Lan and Elyas were used as proof that Warders keep their powers, but Lan had a new bond with Myrelle, so he never actually lost his "Warderness," and Elyas is a Wolf Brother, thus he would have most of those powers anyway.