This needs testing.

The Falling Down/Down State bugs afflict your companions. When afflicted, they either remain in the down state, as though they had suffered excessive damage, or they cycle constantly into the down state, remaining on their feet for a few seconds before falling back down again.

According to wikis and the patch notes, this is believed to be caused by radiation damage.

My own experience showed the symptoms expressing themselves only when Piper was dismissed or replaced as a companion, at which point she would be begin cycling into the down state. This prevented the conversation with Nick at the conclusion of "Reunions," thus breaking my playthrough and preventing any story progress.

In the hopes that a new patch would fix it, I kept wandering around with Piper, figuring that I'd start a new game over if the upcoming patch didn't address the issue. After many days of wandering around, it appears that Piper has been cured. I successfully dismissed her without the bug cycling her constantly into the "down state." I have a theory as to how it may have happened. It is my hope that people suffering from this bug can test the potential solution and let me know if it is, indeed, a fix.

Bring the companion suffering from the bug to Hallucigen.

You may want to grab the related mission from the hotel in Goodneighbour first, but it shouldn't be necessary. The final section of Hallucigen contains a decontamination corridor that removes all radiation damage. It may require three levels of the Hacking perk to activate it, I never noticed any alternative. Send the companion into the decontamination hallway or have them follow you through it.

I don't believe I saved inside Hallucigen, and it was about a 30 hours of playtime ago, so I have no way to test if this works. It seems the most likely explanation, based on the hours I've replayed so far.