I guess this theory is an off shoot of the timeline. It all depends on the accuracy of the dates for the rodeo when Jack met Loreen, and the hospital scene.
I have to review the timeline before having an opinion on its possibility.
In any case, true or not, I don't see it having an impact on the main theme of the movie, the love between Ennis and Jack.
Having said that, it is interesting to speculate what impact there may be, if it were true, that Bobby is not Jack's biological son.
These ideas occurred to me while thinking about it. Presented here in no particular order. Call it a stream of consciousness if you will.
It may explain Jack's standing up to old man Newsome at the Thanksgiving dinner and the old man's treatment of Jack at the hospital. Rodeo was just a stand in, and the baby has no part of Jack, so he looked like a splitting image of grandpa instead of pa. Jack couldn't argue then, but when the old man went too far at the Thanksgiving dinner, implying Bobby watching football was being a real man and Jack wasn't, he said something that could be interpreted as a threat, "my house, ... my son...sit down.." In other words, any back talk from you and I am going to tell "my son" about his real daddy.
It may also explain Jack's initial proposal to Ennis, about old Newsome would be happy to pay him off to go away, now that they are married, and his willingness to leave Loreen and "his son".
However, it makes one wonder about Jack's reluctance to leave Texas, and his seemingly close relationship with Bobby, teaching his to drive a tractor, caring about his education, etc. It is possible that with time, Jack learned to love Bobby like his own, and felt a bond that made it difficult to leave Texas.

In any case, I am not sure if Annie, Diana, Larry, or Ang Lee, intended this to be a significant part of the plot. On the other hand, if Bobby was the reason why Jack didn't move back to Lightening Flat, and made it easier for Ennis to have a sweet life with him, it would be VERY significant. On the other hand, (to the tune of Fiddler on the Roof) biological or not, it is Jack's love for Bobby, which kept him from his sweet life with Ennis, if that were the reason for not leaving Texas.
Would Jack feel different for Bobby if he were not his biological son? I doubt it. The way he told Alma...I have a little boy...is filled with pride and love. In a way, Ennis wasn't the only one with conflicting emotions in the movie.