Interesting though, I don't think I've ever seen a code 507 in the wild!

Does it mean that the logs got so big that the httpd wouldn't serve requests, for want of space to store the record?

I don't know actually. 507 was introduced in rfc 4918 alongside WebDAV, so my naive assumption is that its main use case was for something more associated with authoring or general file manipulation.

I'm surprised that it has much use for a normal GET request, but I guess anything can be a bottleneck!