So here my theory is:
Either A.) Neal Caffrey is Bryce Larkin. So.. Bryce went on trips to New York when he was in college, and ended up meeting Moz.. and became a thief. Then, when "Neal" is everywhere in Europe it's all being covered up because The CIA is covering up Bryces work, meaning they deleted all of "Neal's" video footage on accident. Then, after he is caught and thrown in jail, the CIA pulled him out. Enter all that happened in Chuck. Then, instead of dying in the Intersect room, Beckman ends up reviving him, and he becomes "Neal" to keep him safe.
OR
B.) Neal Caffrey is Bryce Larkin. So.. Bryce went on trips to New York when he was in college, and ended up meeting Moz.. and became a thief. Then, when "Neal" is everywhere in Europe it's all being covered up because The CIA is covering up Bryces work, meaning they deleted all of "Neal's" video footage on accident. Then, after he is caught and thrown in jail, the CIA pulled him out. Enter all that happened in Chuck. Bryce gets revived (somehow) and runs off. Nobody from the Chuck timeline is in the know that he is alive, and he becomes Neal Caffrey.
OR
C.) Neal Caffrey and Bryce Larkin are twins. They were formally known as Danny and Bryce Bennett. The went into WitSec. Then their mom got married to a "Mr. Larkin." Neal finds out about the truth about his real dad (James Bennett-dirty cop), and runs off, changing his name to Neal Caffrey. Boom. Two story lines, two shows, and yet still connected. The reason their birth dates are not the same is because 1. NEAL IS A CONMAN! And 2. BRYCE IS A CIA AGENT! They wouldn't keep their real birthdays! Especially if Neal didn't want to deal with his past as Neal Bennett.
OR
D.) Neal Caffrey and Bryce Larkin are siblings. Same as above, but don't worry about birthdays. Neal is James Bennett's son, but instead of going into WitSec at 2, he went in at 5 - So Bryce could be born from the same parents.