I have several Google Apps (Free Edition) and one traditional Gmail acct set into my Nexus 4 (stock 4.2.2). All accts use 2FA (2 factor authentication).
Using the gmail app, all (GApps & Gmail) but two (both GApps) of those accounts are able to retrieve/sync emails successfully. The settings on the Forwarding and POP/IMAP are identical for all accounts.
I've tried removing the accts, force stop & clear data on gmail app, then recreating both accounts. I've tried leaving (for long periods of time) the auto-sync checkbox ON for those accts. I tried the fix outlined in this video (Auto-Sync data was already enabled, but I disabled, exited, then re-enabled).
What else can I try to get those two accounts working!?