Unless something changed in the past few months, its still possible - I'm at max Machine rep.
i've had a similar problem. i was at 105 zion rep a few months ago. Then i noticed later that same day, that i was down to 102 rep. (after just doing archives). Then finally it went up. Went up to 107 after just doing 1 single standard assassination mission! but since then i still havent been able to raise it any higher. Plus i have only got one "meeting with Niobe" mission, and that was the first one you recieve. i havent had one in 90 or so Rep points. I've tried every district, but it still doesnt show up! ?
Stuck at 126 for years. Missions hate me, so I don't care about it anymore.
Your character isnt' broken, the chance of getting the last few points of rep to the max is very very very small.