Is your 1 companion a human (or mutant or ghoul) ?

You're allowed 2 companions if and only if one is a robot (ED-E or Rex)

Also, did you ever have a human companion? If so, did you forget them somewhere? Outside Caeser's tent, or in the Lucky 38 lobby maybe.

--After recruiting him then attempting to recruit him while you have another team member, might make the dialog to recruit him without another companion vanish. (ie; if you talk to him at the lucky 38 without a partner after trying to recruit him with one, you will not see the option [I think we should travel together] anymore.) To fix this glitch, remove both companions from your group (both humanoid and non-humanoid) then the option should appear again.

--The above fix may not always work, as he will say something along the lines of "You have quite the menagerie at your disposal", and the game will say that you can only have two companions at a time. The only fix currently is to load a previous save (happened after recruiting Arcade, then immediately firing him to get access to a previous companion's inventory, then trying to recruit again). However, before all hope is lost, try talking to him with ED-E, Rex, and neither. Because he has different dialogue options for each situation, one of them might be triggered (confirmed fix multiple times by adding Rex).