It appears that some links are broken during the copy/paste operation.

1. Any effects come across OK and are saved properly.
2. If the audio was linked to video, that link is lost, even though both parts come across OK initially.
3. Upon saving/re-opening the project, the audio is missing from the clip in the sequence, but the representation of the clip is still there.
4. The audio for the clip is still functioning in the Project panel.

Here's the workaround:

1. In the new project, make sure that the video track that contains the missing audio is targeted.
2. Press the "T" (or "M" in CS3) key to do a match frame and load the audio into the Source Monitor.
3. Voila! The audio has returned to the sequence.

Although I consider it to be a minor bug, Adobe may disagree with me. To that end, you should file a bug report here on the Adobe site with your exact reproducible steps and you may include my workaround if you want.

In the future, I recommend importing one project into another to get what you need instead of doing a copy/paste between open and closed projects. Importation of projects is the mechanism that is designed to get objects from one project to another.
Essentially, any time I made an "L" cut using audio from a different video clip than it was originally paired with, the audio clips would disappear from the timeline every time I reopened the project for editing, no matter how many times I saved it.

I have not found a solution yet, but I may be on to something. And it might be ridiculously simple. I am working on a project using the same exact format tapes, same encoder, same hard drive, etc. etc. and experienced the same problem a few more times.

When I made these edits, I was using the overlay button and matting the new audio onto the old. I have found that if I delete the old audio first, and replace the empty space with the new, the clips don't disappear.

Again I'm not positive it's as simple as that, but once I started doing that, the clips remained. If I experience any more issues with it, I'll post again.
Sometimes I open up a project separate from the one I'm working on in order to copy some timeline elements to paste into the first project. Often the audio waveform, and the audio itself, disappears while leaving the reference on the timeline. These "unlinkings" to the clips in the bin seem random and I have no explanation, but I do have a workaround:

Right click the timeline reference that lost its link, select Reveal in Project. The clip is then highlighted in the bin. Double click that clip to place it in the source monitor. Now here's the big trick: In the source monitor, simply click on either the Mark In or Mark Out markers. In my world, immediately the waveform reappears within the timeline reference, life is good again!
Community content is available under CC-BY-SA unless otherwise noted.