jpetersen
If you look in your project file you'll see it takes all your selected files, packs them into one big file and normalizes that.
Your individual audio clips are now in fact pointers into that one mega audio file.
I submitted a bug report way back when but it never got fixed.
Save to a new directory with the one audio file per clip setting separates them out again.
Anderton
Add one more key command to bounce each clip to itself first. 
Ok, it seems like a combination of setting the one-audio-file per clip, and bouncing each clip to itself solves the problem.
The behaviour seems to be as follows:
1. Without the one-audio-file-per-clip setting, the whole normalize a clip thing is buggy unless I
(a) Always select the clip with the mouse and;
(b) Only normalize one clip at a time.
2. With one-audio-file-per-clip enabled, but WITHOUT bouncing the individual clips to themselves:
(a) For multi-select, if all the clips are contiguous, then it treats them as a single clip - i.e. the normalize for each clip is affected by the volume in the loudest clip, as if they were indeed just one clip.
(b) For multi-select, if the clips are not contiguous, then the bug returns and it's random as to whether I get a blank or corrupted clip, or a successful normalization.
(c) For single-select, it seems to work fine. So I can do my CTRL+ALT+KeyPad 6, CTRL+ALT+N, ENTER, repeatedly to go through each individual clip.
3. With one-audio-file-per-clip enabled, and WITH bouncing the individual clips to themselves:
(a) I can multi-select all the clips and the normalize seems to apply the effect individually to each clip - i.e. the volume of the other selected clips is not taken into account.
(b) The repeated CTRL+ALT+KeyPad 6, CTRL+ALT+N, ENTER works too, as expected.
Anderton
msmcleod
Even if I drag them over individually, it doesn't retain the original time position of the clips so putting them back together would be a nightmare (none of these clips are snapped to any grid value, because they're generally parts of vocal phrases).
I forgot you need to drag them over individually, so bouncing to itself with a key command might be faster. However, they will retain their time position if you go Edit > Preferences > File > Audio and check both "Export Broadcast Waves by Default" and "Always Import Broadcast Waves at their Timestamp."
Given that I have to go through each clip individually anyhow, I'm not sure doing an export will speed things up here. I will take your suggestion for having a key binding for bouncing though, so I can just repeat this (with CTRL+ALT held down):
- KeyPad 6 (select next clip)
- B (bounce to clip - i.e. bounce this clip to itself)
- N (normalize dialog pops up)
- ENTER (apply normalize)
I'm really dumbfounded as to how inconsistent the behaviour is here though, and I've not tested these scenarios extensively enough to be confident they'll work all the time.
@Noel & the team - Please fix this bug!