From A Wiki of Ice and Fire
It is widely theorized that the "Bastard Letter" sent to Jon Snow, apparently from Ramsay, claiming Stannis is dead and challenging Jon Snow, is false, and may not even be written by Ramsay.
- The letter is signed by Ramsay and sealed with pink wax.
- The rescue attempt was discovered, meaning Mance could be a prisoner and his spear-wives dead.
- The letter is unlike other letters by Ramsay, who previously sent pieces of skin in his letters and used blood to write them.
- The smear of pink is in contrast to the Bolton seal on Ramsay's other letters.
- It is implausible that there could be seven days of battle.
- Ramsay asking for Jeyne Poole and Theon is odd, considering when last heard of they were in the Crofters Village and are unlikely to be able to escape. Jeyne was sent away, but Theon is imprisoned in the village.
- Jon Snow doesn't know that Ramsay forced the name "Reek" on Theon Greyjoy: Ramsay should know this, and to ask for "Theon" by name. Again, if he really defeated Stannis he would have recaptured Theon, and not asked for him at all. It is still possible that Ramsay wrote the letter but didn't defeat Stannis, but is lying in its contents - and in this case, Ramsay assumes that Theon and Jeyne managed to make the long journey to the Wall on foot, through heavy snows, which is itself unlikely.
- Maybe Ramsay is in such a fury at losing his bride and "his Reek" that he isn't thinking clearly - and he wasn't a very rational character to begin with. So asking Jon for Theon by a name he doesn't know might possibly be a simple oversight on Ramsay's part.
- Stannis may be tricking Jon into bringing assistance.
- His last appearance implied he had a plan to defeat the Freys and Boltons, meaning he was likely not defeated.
- Stannis knows it would take many days for Jon to come from the Wall.
- Jon Snow has no way of knowing that "Reek" refers to Theon Greyjoy. Whoever wrote the letter honestly believes that Theon has escaped to the Wall, and thus could explain to Jon that Ramsay forced the name "Reek" on him. If Stannis still has Theon in his custody he would know this, and wouldn't have used a name Jon wouldn't recognize.
- Is Stannis really the type to make Jon break his vows by "tricking" him?
- He is probably literate, as shown by the Abel anagram.
- Mance may be attempting to bring the Wildlings south, knowing they will be loyal to him.
- The message may contain a code for Melissandre, the letter even says "Tell his red whore."
- The words false King and Black Crows match with the Wildling style of speaking.
- When last seen, the rescue attempt was going wrong, meaning Mance may not have been in a position to send letters.
- Wyman Manderly is actively planning to betray the Boltons.
- Wyman, as a northman, would understand better than southerners that the Night's Watch (and the thousands of wildlings it is sheltering) wouldn't move against the Boltons due to their political neutrality. Wyman may therefore have wanted to frame Ramsay for threatening to murder the Lord Commander of the Night's Watch, to provide Jon Snow with a legal excuse to finally move against the Boltons.
- Stannis may have in fact been defeated but forced a pyrrhic victory on the Boltons. Wyman would know this because he's with the Boltons at Winterfell. In which case this would spur his decision to get Jon Snow and the Night's Watch to help.
- Wyman has been present with the Boltons at Winterfell, and thus would probably be somewhat familiar with Ramsay's speaking/writing style: he knows how insulted Ramsay gets when called a bastard, and would know that a letter Ramsay wrote to Jon would be filled with insults about his bastard status.
- Being present at Winterfell, Wyman would know that exactly six spearwives were killed, and if Ramsay captured Mance and put him on public display.
- Wyman has demonstrated his use of subterfuge before: making a show of dragging off Davos for execution to convince the Boltons/Lannisters of his loyalty, when his true intentions were actually the opposite. Stannis is willing to use ruses of war against his enemies, but is typically more honest with his allies: Wyman, in contrast, has been shown to be willing to trick his allies if it will serve the greater good, i.e. not tell Davos his real intentions so Davos's fear at being dragged away would seem genuine to the Freys.
- Wyman doesn't know that Stannis has Theon and Jeyne Poole, so he may have falsely assumed that "Arya Stark" would flee to the Wall. Wyman was at Winterfell and knows that Ramsay calls Theon "Reek". If Wyman believes that Theon and Jeyne managed to get to the Wall, he would use the name "Reek" to convincingly imitate Ramsay, and assume Theon would explain it to him.
- It isn't clear how Ramsay would even know about Mance's infant son and sister-in-law Val at the Wall (unless he tortured the information out of him, but he wouldn't know to ask in the first place). Davos, however, was at the Wall with Stannis, and would have known about what prisoners were taken after the Battle of Castle Black (even if Davos stayed at Eastwatch-by-the-Sea, Stannis would have given him a report from Castle Black in the same written message that instructed him to go to White Harbor). Davos, in turn, could have told Wyman Manderly once he revealed his true plans.
- Jon Snow had to directly explain to Stannis that the position of King-Beyond-the-Wall is not hereditary, and thus Mance's son isn't really a "prince" nor his sister-in-law a "wildling princess". In contrast, the Manderlys are located at the southernmost part of the North, and rarely have to interact with wildlings - perhaps enough that Wyman himself doesn't know this detail about wildling culture.
- When Jon Snow reads the letter, he makes no mention that it isn't in Ramsay's handwriting.
- Ramsay could have found out about "the wildling princess" and the "little prince" by just torturing one of Stannis's footsoldiers or scouts he managed to capture, who similarly wouldn't understand that the wildlings wouldn't use those terms.