I get a lot of broken images. Almost every image that needs resized, comes out broken. It does so after about 2s of processing something. And then the next one (fails too), and the next, and the next, and so on.

On my server, I can clearly see the apache2 processing sweating like a horse to (presumably) resize those images. So it's doing something, but what?

At the time of writing, my server has over 2.5GB of completely unused memory. Also it has enough horsepowers, and plenty of free disk space. It's also set up to use imagemagick.

PHP related config:
max_execution_time 120
max_input_time 120
memory_limit 500M

Server details:
Apache 2.4.18
PHP 5.6.18
MySQL 5.5.47
ImageMagick 6.7.7-10
Zenphoto 1.4.9

To answer my own question: permission problems.

Brilliant, somehow part of the cache had gotten the wrong owner/group and became inaccessible for apache.

I'll keep an eye on it, see if it happens again.

Same problem - how do you fix that?
I put up 2 albums, one is fine, one has random broken images. I made them smaller and re-uploaded. No luck.
I don't see the images in the Albums/Images tab, but if I click on them in the online gallery they open as an image on a new page - but outside of the zenphoto theme. So I know they're up there and can be opened.

I'm on godaddy so shouldn't be server problems.

First review your server PHP error log about any possible issue.
Then please read these two documents:
http://www.zenphoto.org/news/problems-with-albums-and-images
http://www.zenphoto.org/news/permissions-for-zenphoto-files-and-folders