I tested by reloading to before the bug expressed itself, but the bug is retroactive and permanent, and cannot be avoided in my game.

This is how the bug came about:

- I had Nick Valentine as my current companion. I had previously completed all stages of "The Dig" except the final one in which I would speak to Hancock.

- After finishing a main quest mission, I stopped by the State House and completed the final stage by speaking to Hancock.

- After speaking to Hancock, I followed him with Nick out onto the balcony for his speech.

- After his speech, I spoke to him and proceeded to complete the recruitment.

- I sent Nick to Sanctuary Hills the first and second times I tested, and I sent him to County Crossing the third time. In all three cases, Nick permanently disappeared from the game, and cannot be recovered. He no longer spawns at any location, including his home. All items he was carrying disappeared as well.

I do not know what factors are at play. I assumed that a hidden population limit had broken him at Sanctuary Hills, where I had a large number of settlers and companions, but the bug occurred regardless of whether he was sent to that location.

Edit: The bug is applied to whichever one of the two is not my companion as soon as I leave that scene.

I tried reloading and not talking to Hancock, and Hancock entered the State House and was permanently deleted from the game. Again, he would not appear at any locations, including Goodneighbour. It would appear that there is an unintended cap on the number of companions, and that any number beyond that cap are permanently deleted.

Edit 2: For some reason, the bug seems related to whether the "dismissed" companion is sent anywhere specific, or whether the specific location is a settlement with an existing population. If not sent anywhere specific, or if they are sent to a settlement with an existing population, they are permanently deleted. This occurs as soon as either of those happens, and is permanent. The temporary workaround seems to be to constantly dismiss the companions to unpopulated settlements. Obviously, this is an extremely limited workaround that will not last for long, and eventually, a companion is going to be permanently deleted. It may well occur at any time while I am wandering around, and there will be no warning, meaning that I won't likely discover which companion has been permanently deleted until long after the fact, when I go looking for that companion.

I'm having the same issue where I went to switch from Piper to Valentine, and when I was to select Piper's destination I canceled, but it said it sent her to Sanctuary, I believe. After finishing my quest and going back to Sanctuary to look for Piper, she is totally gone. More details of what I've tried here: http://www.gamesas.com/topic/1567284-piper-has-been-missing-for-a-really-long-time/