It's back to not syncing...
I had talked with an Apple customer service person, he told me to sign out of iCloud (from my Mac) and sign back in. I did that and my Mac was able to sync with iCloud again (specifically Numbers documents). Well, that only lasted a day or so, it's not syncing again. I tried signing out and back in again, this time it did not fix it.
My iPad seems to sync Numbers documents with iCloud just fine (I was able to confirm by visiting icloud.com and seeing the modify date & time of the Numbers doc), so the problem seems to be only with my Mac. (Mac Pro, 10.8.2.)
When I open Numbers, load a document from iCloud (but it's really just in cache locally), then make a change and try to save, this is everything that comes up in console:
1.21.13 2:57:21.837 PM librariand[391]: item update observer error: Connection invalid
1.21.13 2:57:36.417 PM librariand[391]: item update observer error: Connection invalid
1.21.13 2:58:46.191 PM Numbers[16241]: Object (0x566f0b0 of class __NSDictionaryI) named "NSMutableDictionary-0" was already registered with another name "NSMutableDictionary-2".
1.21.13 2:58:46.191 PM Numbers[16241]: Object (0x566f0b0 of class __NSDictionaryI) named "NSMutableDictionary-0" was already registered with another name "NSMutableDictionary-3".
1.21.13 2:58:46.194 PM Numbers[16241]: Object (0x566f0b0 of class __NSDictionaryI) named "NSMutableDictionary-0" was already registered with another name "NSMutableDictionary-4".
1.21.13 2:58:46.205 PM Numbers[16241]: Object (0x566f0b0 of class __NSDictionaryI) named "NSMutableDictionary-0" was already registered with another name "NSDictionary-0".
1.21.13 2:58:46.236 PM Numbers[16241]: NameKeyHashSet and ObjectHashMap out of sync: object map entry for __NSDictionaryI 0x566f0b0 references name "NSMutableDictionary-0" instead of "NSMutableDictionary-2"
1.21.13 2:58:46.236 PM Numbers[16241]: NameKeyHashSet and ObjectHashMap out of sync: object map entry for __NSDictionaryI 0x566f0b0 references name "NSMutableDictionary-0" instead of "NSDictionary-0"
1.21.13 2:58:46.236 PM Numbers[16241]: NameKeyHashSet and ObjectHashMap out of sync: object map entry for __NSDictionaryI 0x566f0b0 references name "NSMutableDictionary-0" instead of "NSMutableDictionary-3"
1.21.13 2:58:46.237 PM Numbers[16241]: NameKeyHashSet and ObjectHashMap out of sync: object map entry for __NSDictionaryI 0x566f0b0 references name "NSMutableDictionary-0" instead of "NSMutableDictionary-4"
1.21.13 2:58:54.397 PM librariand[391]: item update observer error: Connection invalid
I'm not really sure what's going on. I do have little snitch, but I've turned it completely off for tests, so that's not interfering. I'll wait a day or two, see if any great ideas pop up, then I'll call Apple tech support again.