I have recently (since ~1month ago) experienced the following problem:
When I get new mail on my gmail account, Gmail will usually not autimatically retrieve it (e.g., no notification; and when I manually open GMail app, the mail will NOT be in the inbox)
However, manual mail retrieval/sync DOES work fine, for example if I click on "Sync" (retreieve) curcle icon within GMail app; or on Elixir 2's "start background data sync" icon; or check-off-and-back-on on the "Mail Sync" in Settings->Accounts for relevant Google account.
Also, sometimes Gmail does get retrieved/notified; I am not 100% sure but there seems to be a very strong correllation between that and having recently opened GMail app (especially in foreground).
GMail sync worked perfectly find till ~1 month ago
The background sync does not work for ANY of the 3 Gmail accounts I registered.
OTHER mail apps seem to sync perfectly fine (e.g. Hotmail or Yahoo mail) in the background.. So I have a hard time believing it is a problem with overall sync.
The issue is independent of network - it doesn't work on WiFi or 3G.
Droid X, running rooted stock Froyo 2.2.
I have gone through 100% of intents in Autostarts and made sure every single intent is ON for both Gmail listener and Google Framework.
Gmail version is 2.3.6 (it has a 7/1/2012 backup in Titanium Backup Pro, so I'm assuming this is when it was last updated. BUT the problem existed before 7/1)
Background sync is turned ON - confirmed via Quick Settings app as well as widgets from Elixir 2. (also, as noted above, Hotmail and Yahoo mail seem to sync fine)